I’ve ended up spending quite a lot of time researching premodern economic growth, as part of a hobby project that got out of hand. I’m sharing an informal but long write-up of my findings here, since I think they may be relevant to other longtermist researchers and I am unlikely to write anything more polished in the near future. Click here for the Google document.
Over the next several centuries, is the economic growth rate likely to remain steady, radically increase, or decline back toward zero? This question has some bearing on almost every long-run challenge facing the world, from climate change to great power competition to risks from AI.
One way to approach the question is to consider the long-run history of economic growth. I decided to investigate the Hyperbolic Growth Hypothesis: the claim that, from at least the start of the Neolithic Revolution up until the 20th century, the economic growth rate has tended to rise in proportion with the size of the global economy. This claim is made in a classic 1993 paper by Michael Kremer. Beyond influencing other work in economic growth theory, it has also recently attracted significant attention within the longtermist community, where it is typically regarded as evidence in favor of further acceleration. An especially notable property of the hypothesized growth trend is that, if it had continued without pause, it would have produced infinite growth rates in the early twenty-first century.
I spent time exploring several different datasets that can be used to estimate pre-modern growth rates. This included a number of recent archeological datasets that, I believe, have not previously been analyzed by economists. I wanted to evaluate both: (a) how empirically well-grounded these estimates are and (b) how clearly these estimates display the hypothesized pattern of growth.
Ultimately, I found very little empirical support for the Hyperbolic Growth Hypothesis. While we can confidently say that the economic growth rate did increase over the centuries surrounding the Industrial Revolution, there is approximately nothing to suggest that this increase was the continuation of a long-standing hyperbolic trend. The alternative hypothesis that the modern increase in growth rates constituted a one-off transition event is at least as consistent with the evidence.
The premodern growth data we have is mostly extremely unreliable: For example, so far as I can tell, Kremer’s estimates for the period between 10,000BC and 400BC ultimately derive from a single speculative paragraph in a book published decades earlier. Putting aside issues of reliability, the various estimates I considered also, for the most part, do not clearly indicate that pre-modern growth was hyperbolic. The most empirically well-grounded datasets we have are at least weakly in tension with the hypothesis. Overall, though, I think we are in a state of significant ignorance about pre-modern growth rates.
Beyond evaluating these datasets, I also spent some time considering the growth model that Kremer uses to explain and support the Hyperbolic Growth Hypothesis. One finding is that if we use more recent data to estimate a key model parameter, the model may no longer predict hyperbolic growth: the estimation method that we use matters. Another finding, based on some shallow reading on the history of agriculture, is that the model likely overstates the role of innovation in driving pre-modern growth.
Ultimately, I think we have less reason to anticipate a future explosion in the growth rate than might otherwise be supposed.
EDIT: See also this addendum comment for an explanation of why I think the alternative "phase transition" interpretation of the Industrial Revolution is plausible.
Thank you to Paul Christiano, David Roodman, Will MacAskill, Scott Alexander, Matt van der Merwe, and, especially, Asya Bergal for helpful comments on an earlier version of the document. ↩︎
By "economic growth rate," here, I mean the growth rate of total output, rather than the growth rate of output-per-person. ↩︎
As one example, which includes a particularly clear summary of the hypothesis, see this Slate Star Codex post. ↩︎
I wrote nearly all of this document before the publication of David Roodman’s recent Open Philanthropy report on long-run economic growth. That report, which I strongly recommend to anyone interested in long-run growth, has some overlap with this document. However, the content is fairly different. First, relative to the report, which makes novel contributions to economic growth modeling, the focus of this doc is more empirical than theoretical. I don’t devote much space to relevant growth models, but I do devote a lot of space to the question: “How well can we actually estimate historical growth rates?” Second, I consider a wider variety of datasets and methods of estimating historical growth rates. Third, for the most part, I am comparing a different pair of hypotheses. The report mostly compares a version of the Hyperbolic Growth Hypothesis with the hypothesis that the economic growth rate has been constant throughout history; I mostly compare the Hyperbolic Growth Hypothesis with the hypothesis that, in the centuries surrounding the Industrial Revolution, there was a kind of step-change in the growth rate. Fourth, my analysis is less mathematically rigorous. ↩︎
There is also ongoing work by Alex Lintz to analyze available archeological datasets far more rigorously than I do in this document. You should keep an eye out for this work, which will likely supersede most of what I write about the archeological datasets here. You can also reach out to him (firstname.lastname@example.org) if you are interested in seeing or discussing preliminary findings. ↩︎
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:
Here's how I would summarize the evidence in your document:
Here's my take in more detail:
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.
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:
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 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.:
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.
I'm going to try and restate what's going on here, and I want someone to tell me if it sounds right:
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.
Hi there - Anton Howes here, an economic historian. My speciality is the Industrial Revolution, and especially the causes of accelerating invention. Stephen Clare emailed me about this debate in the comments, and asked for my take.
A few things that I don't think have been mentioned, or perhaps might do with being made a little more explicit (though apologies if I missed them - has been a lot to catch up on reading through):
In the linked doc, I mainly contrast two different perspectives on the Industrial Revolution.
Stable Dynamics: The core dynamics of economic growth were stable between the Neolithic Revolution and the 20th century. Growth rates increased substantially around the Industrial Revolution, but this increase was nothing new. In fact, growth rates were generally increasing throughout this lengthy period (albeit in a stochastic fashion). The most likely cause for the upward trend in growth rates was rising population levels: larger populations could come up with larger numbers of new ideas for how to increase economic output.
Phase Transition: The dynamics of growth changed over the course of the Industrial Revolution. There was some barrier to growth that was removed, some tipping point that was reached, or some new feedback loop that was introduced. There was a relatively brief phase change from a slow-growth economy to a fast-growth economy. The causes of this phase transition are somewhat ambiguous.
In the doc, I essentially argue that existing data on long-run growth doesn’t support the “stable dynamics” perspective over the “phase transition” perspective. I think that more than anything else, due to data quality issues, we are in a state of empirical ignorance.
I don’t really say anything, though, about the other reasons people might have for finding one perspective more plausible than the other. Since I personally lean toward the “phase change” perspective, despite its relative inelegance and vagueness, I thought it might also be useful for me to write up a more detailed comment explaining my sympathy for it.
Here, I think, are some points that count in favor of the phase change perspective.
1. So far as I can tell, most prominent economic historians favor the phase change perspective.
For example, here is Joel Mokyr describing his version of the phase change perspective (quote stitched together from two different sources):
And here’s Robert Allen telling another phase transition story (quotes stitched together from Global Economic History: A Very Short Introduction):
I’m not widely read in this area, but I don’t think I’ve encountered any prominent economic historians who favor the “Stable Dynamics” perspective (although some growth theorists appear to).
2. The stable dynamics perspective is in tension with the extremely “local” nature of the Industrial Revolution.
Although a number of different countries were experiencing efflorescences in the early modern period, the Industrial Revolution was a pretty distinctly British (or, more generously, Northern European) phenomenon. An extremely disproportionate fraction of the key innovations were produced within Britain. During the same time period, for example, China is typically thought to have experienced only negligible technological progress (despite being similarly ‘advanced’ and having something like 30x more people). Economic historians also typically express strong skepticism that any country other than Britain (or at best its close neighbors) was moving toward an imminent industrial revolution of its own. See, for example, the passages I quote in this comment on the economy of early modern China.
This observation fits decently well with phase transition stories, such as Robert Allen’s: the British economy achieved ignition, then the fire spread to other states. The observation seems to fit less well, though, with the “stable dynamics” perspective. Why should the Industrial Revolution have happened in a very specific place, which held only a tiny portion of the world’s population and which was until recently only an economic ‘backwater’?
Mokyr expresses skepticism on similar grounds (p. 36-37).
3. There has been vast cross-country variation in growth rates, which isn’t explained by differences in scale
In modern times, there are many examples of countries that have experienced consistently low growth rates relative to others. This suggests that there can be fairly persistent barriers to growth, other than insufficient scale, which cause growth rates to be substantially lower than they otherwise would be. As an extreme example, South Korea’s GDP growth rate may have been about an order-of-magnitude higher than North Korea’s for much of its history: despite many other similarities, institutional barriers were sufficient to keep North Korea’s growth rate far lower. (The start of South Korea’s “growth miracle” also seems like it could be pretty naturally described as a phase transition.)
At least in principle, it seems plausible that some barriers to growth -- institutional, cultural, or material -- affected all countries before the Industrial Revolution but only affected some afterward. Along a number of dimensions, states that are growing quickly today used to be a lot more similar to states that are growing slowly today. They also faced a number of barriers to growth (e.g. the need to rely entirely on ‘organic’ sources of energy; the inability to copy or attract investments from ultra-wealthy countries; etc.) that even the poorest countries typically don’t have today.
Acemoglu makes a similar point, in his growth textbook, when talking about the Kremer model (p. 114):
4. It’s not too hard to develop formal growth models that exhibit phase transitions
For example, there are models that formalize Robert Allen’s theory, “two sector” models in which the industrial sector overtakes the agricultural sector (and causes the growth rate to increase) once a certain level of technological maturity is reached, models in which physical capital and human capital are complementary (and a shock that increases capital-per-worker makes it rational to start investing in human capital), and models in which insufficient property protections limit the rate of growth (by capping incentives to innovate and invest). For example, here’s a classic two sector model.
I don’t necessarily “buy” any of these specific models, but they do suffice to show that there are a number of different ways you could potentially get phase transitions in economic growth processes.
5. The key forces driving and constraining post-industrial growth seem fairly different from the key forces driving and constraining pre-industrial growth
Technological and (especially) scientific progress, or what Mokyr calls “the growth of useful knowledge,” seems to play a much larger role in driving post-industrial growth than it did in driving pre-industrial growth. For example, based on my memory of the book The Economic History of China, a really large portion of China’s economic growth between 200AD and 1800AD seems to be attributed to new crops (first early ripening rice from Champa, then American crops); to land reclamation (e.g. turning marshes into rice paddies; terracing hills; planting American crops where rice and wheat wouldn’t grow); and to the more efficient allocation of resources (through expanding markets or changes in property rights). The development or improvement of machines, or even the development or improvement of agricultural and manufacturing practices, doesn’t seem to have been a comparably big deal. The big growth surge that both Europe and China experienced in the early modern period, and which may have partly set Britain for its Industrial Revolution, also seems to be mostly a matter of market expansion and new crops.
For example, Mokyr again:
There are also a couple obvious material constraints that apply much more strongly in pre-industrial than post-industrial societies. First, agricultural production is limited by the supply of fertile land in a way that industrial production (or the production of services) is not; if you double capital and labor, without doubling land, agricultural production will tend to exhibit more sharply diminishing returns.
Second, and probably more importantly, pre-industrial economic production relies almost entirely on ‘organic’ sources of energy. If you want to make something, or move something, then the necessary energy will typically come from: (a) you eating plants, (b) you feeding plants to an animal, or (c) you burning plants. Wind and water can also be used, but you have no way of transporting or storing the energy produced; you can’t, for example, use the energy from a waterwheel to power something that’s not right next to the waterwheel. This all makes it just really, really hard to increase the amount of energy used per person beyond a certain level. Transitioning away from ‘organic’ sources of energy to fossil fuels, and introducing means of storing/transmitting/transforming energy, intuitively seems to remove a kind of soft ceiling on growth. (Some people who have made a version of this point are: Vaclav Smil, Ian Morris, John Landers, and Jack Goldstone. It's also sort of implicit in Robert Allen's model.) It’s especially notable that, for all but the most developed countries, total energy consumption within a state tends to be fairly closely associated with total economic output.
To be clear, this super long addendum has only focused on reasons to take the “phase transition hypothesis" seriously. I’ve only presented one side. But I thought it might still be useful to do this, since the reasons to take the “phase transition perspective” seriously are probably less obvious than the reasons to take the “constant dynamics perspective” seriously.
Of course, my descriptions of these two perspectives are far from mathematically precise. There is some ambiguity about what it means for one perspective to be “more true” than the other. This paper by Chad Jones, for example, describes a model that combines bits of the two perspectives. ↩︎
As another point of clarification, growth theory work in this vein does tend to suggest that important changes happened during the nineteenth century: once productivity growth becomes fast enough, and people start to leave the Malthusian state, certain new dynamics come into play. However, the high rate of growth in the nineteenth century is understood to result from growth dynamics that have been essentially stable since early human history. ↩︎
One version of the phase change model that I think is worth highlighting: S-curve growth.
Basically, the set of transformative innovations is finite, and we discovered most of them over the past 200 years. Hence, the Industrial Revolution was a period of fast technological growth, but that growth will end as we run out of innovations.The hockey-stick graph will level out and become an S-curve, as g→0.
I think your confusion with the genetics papers is because they are talking about _effective_ population size (N~e~), which is not at all close to 'total population size'. Effective population size is a highly technical genetic statistic which has little to do with total population size except under conditions which definitely do not obtain for humans. It's vastly smaller for humans (such as 10^4) because populations have expanded so much, there are various demographic bottlenecks, and reproductive patterns have changed a great deal. It's entirely possible for effective population size to drop drastically even as the total population is growing rapidly. (For example, if one tribe with new technology genocided a distant tribe and replaced it; the total population might be growing rapidly due to the new tribe's superior agriculture, but the effective population size would have just shrunk drastically as a lot of genetic diversity gets wiped out. Ancient DNA studies indicate there has been an awful lot of population replacements going on during human history, and this is why effective population size has dropped so much.) I don't think you can get anything useful out of effective population size numbers for economics purposes without making so many assumptions and simplifications as to render the estimates far more misleading than whatever direct estimates you're trying to correct; they just measure something irrelevant but misleadingly similar sounding to what you want.
Thanks for the clarifying comment!
I'd hoped that effective population size growth rates might be at-least-not-completely-terrible proxies for absolute population size growth rates. If I remember correctly, some of these papers do present their results as suggesting changes in absolute population size, but I think you're most likely right: the relevant datasets probably can't give us meaningful insight into absolute population growth trends.
Thank you Ben for this thoughtful and provocative review. As you know I inserted a bunch of comments on the Google doc. I've skimmed the dialog between you and Paul but haven't absorbed all its details. I think I mostly agree with Paul. I'll distill a few thoughts here.
1. The value of outside views
In a previous comment, Ben wrote:
Kahneman and Tversky showed that incorporating perspectives that neglect inside information (in this case the historical specifics of growth accelerations) can reduce our ignorance about the future--at least, the immediate future. This practice can improve foresight both formally--leading experts to take weighted averages of predictions based on inside and outside views--and informally--through the productive friction that occurs when people are challenged to reexamine assumptions. So while I think the feeling expressed in the quote is understandable, it's also useful to challenge it.
Warning label: I think it's best not to take the inside-outside distinction too seriously as a dichotomy, nor even as a spectrum. Both the "hyperbolic" and the sum-of-exponentials models are arguably outside views. Views based on the growth patterns of bacteria populations might also be considered outside views. Etc. So I don't want to trap myself or anyone else into discussion about which views are outside ones, or more outsiderish. When we reason as perfect Bayesians (which we never do) we can update from all perspectives, however labeled or categorized.
2. On the statement of the Hyperbolic Growth Hypothesis
The current draft states the HGH as
I think this statement would be more useful if made less precise in one respect and more precise in another. I'll explain first about what I perceive as its problematic precision.
In my paper I write a growth equation more or less as g_y = s * y ^ B where g_y is the growth rate of population or gross world product and ^ means exponentiation. If B = 0, then growth is exponential. If B = 1, then growth is proportional to the level, as in the HGH definition just above. In my reading, Ben's paper focuses on testing (and ultimately rejecting) B = 1. I understand that one reason for this focus is that Kremer (1993) finds B = 1 for population history (as does von Foerster et al. (1960) though that paper is not mentioned).
But I think the important question is not whether B = 1 but whether B > 0. For if 0 < B < 1, growth is still superexponential and y still hits a singularity if projected forward. E.g., I estimate B = ~0.55 for GWP since 10,000 BCE. The B > 0 question is what connects most directly to the title of this post, "Does Economic History Point Toward a Singularity?" And as far as I can see a focus on whether B = 1 is immaterial to the substantive issue being debated in these comments, such as whether a model with episodic growth changes is better than one without. If we are focusing on whether B = 1, seemingly a better title for this post would be "Was Kremer (1993) wrong?"
To be clear, the paper seems to shift between two definitions of hyperbolic growth: usually it's B = 1 ("proportional"), but in places it's B > 0. I think the paper could easily be misunderstood to be rejecting B > 0 (superexponential growth/singularity in general) in places where it's actually rejecting B = 1 (superexponential growth/singularity with a particular speed). This is the sense in which I'd prefer less specificity in the statement of the hyperbolic growth hypothesis.
I'll explain where I'd ideally want more specificity in the next item.
3. The value of an explicit statistical model
We all recognize that the data are noisy, so that the only perfect model for any given series will have as many parameters as data points. What we're after is a model that strikes a satisfying balance between parsimony (few parameters) and quality of fit. Accepting that, the question immediately arises: how do you measure quality of fit? This question rarely gets addressed systematically--not in Ben's paper, not in the comments on this post, not in Hanson (2000), nor nearly all the rest of the literature. In fact Kremer (1993) is the only previous paper I've found that does proper econometrics--that's explicit about its statistical model, as well as the methods used to fit it to data, the quality of fit, and validity of underlying assumptions such as independence of successive error terms.
And even Kremer's model is not internally consistent because it doesn't take into account how shocks in each decade, say, feed into the growth process to shape the probability distribution for growth over a century. That observation was the starting point for my own incremental contribution.
To be more concrete, look back at the qualifiers in the HGH statement: "tended to be roughly proportional." Is the HGH, so stated, falsifiable? Or, more realistically, can it be assigned a p value? I think the answer is no, because there is no explicitly hypothesized, stochastic data generating process. The same can be asked of many statements in these comments, when people say a particular kind of model seems to fit history more or less well. It's not fully clear what "better" would mean, nor what kind of data could falsify or strongly challenge any particular statement about goodness of fit.
I don't want to be read as perfectionist about this. It's really hard in this context to state a coherent, rigorously testable statistical model: the quantity of equations in my paper is proof. And at the end of the day, the data are so bad that it's not obvious that fancy math gives us more insight than hand-wavy verbal debate.
I would suggest however that is important to understand the conceptual gap, just as we try to incorporate Bayesian thinking into our discourse even if we rarely engage in formal Bayesian updating. So I will elaborate.
Suppose I'm looking at a graph of population over time and want to fit a curve to it. I might declare that the one true model is
y = f(t) + e
where f is exponential or what-have-you, and e is an error term. It is common when talking about long-term population or GWP history to stop there. The problem with stopping there is that every model then fits. I could postulate that f is an S-curve, or the Manhattan skyline in profile, a fractal squiggle, etc. Sure, none of these f 's fit the data perfectly, but I've got my error term e there to absorb the discrepancies. Formally my model fits the data exactly.
The logical flaw is the lack of characterization of e. Classically, we'd assume that all of the values of e are drawn independently from a shared probability distribution that has mean 0 and that is itself independent of t and previous values of y. These assumptions are embedded in standard regression methods, at least when we start quoting standard errors and p values. And these assumptions will be violated by most wrong models. For example, if the best-fit S-curve predicts essentially zero growth after 1950 while population actually keeps climbing, then after 1950 discrepancies between actual and fitted values--our estimates of e--will be systematically positive. They will be observably correlated with each other, not independent. This is why something that sounds technical, checking for serial correlation, can have profound implications for whether a model is structurally correct.
I believe this sort of fallacy is present in the current draft of Ben's paper, where it says, "Kremer’s primary regression results don’t actually tell us anything that we didn’t already know: all they say is that the population growth rate has increased." (emphasis in the original) Kremer in fact checks whether his modeling errors are independent and identically distributed. Leaving aside whether these checks are perfectly reassuring, I think the critique of the regressions is overdrawn. The counterexample developed in the current draft of Ben's paper does not engage with the statistical properties of e.
More generally, without explicit assumptions about the distribution of e, discussions about the quality of various models can get bogged down. For then there is little rigorous sense in which one model is better than another. With such assumptions, we can say that the data are more likely under one model than under another.
4. I'm open to the hyperbolic model being too parsimonious
The possibility that growth accelerates episodically is quite plausible to me. And I'd put significant weight on the the episodes being entirely behind us. In fact my favorite part of Ben's paper is where it gathers radiocarbon-dating research that suggests that "the" agricultural revolution, like the better-measured industrial revolution, brought distinct accelerations in various regions.
In my first attack on modeling long-term growth, I chose to put a lot of work into the simpler hyperbolic model because I saw an opportunity to improve is statistical expression, in particular by modeling how random growth shocks at each moment feed into the growth process and shape the probability distribution for growth over finite periods such as 10 years. Injecting stochasticity into the hyperbolic model seemed potentially useful for two reasons. For one, since adding dynamic stochasticity is hard, it seemed better to do it in a simpler model first.
For another, it allowed a rigorous test of whether second-order effects--the apparently episodic character of growth accelerations--could be parsimoniously viewed as mere noise within a simpler pattern of long-term acceleration. Within the particular structure of my model, the answer was no. For example, after being fit to the GWP data for 10,000 BCE to 1700 CE, my model is surprised at how high GWP was in 1820, assigning that outcome a p value of ~0.1. Ben's paper presents similar findings, graphically.
So, sure, growth accelerations may be best seen as episodic.
But, as noted, it's not clear that stipulating an episodic character should in itself shift one's priors on the possibility of singularity-like developments. Hanson (2000)'s seminal articulation of the episodic view concludes that "From a purely empirical point of view, very large changes are actually to be expected within the next century." He extrapolates from the statistics of past explosions (the few that we know of) to suggest that the next one will have a doubling time of days or weeks. He doesn't pursue the logic further, but could have. The next revolution after that could come within days and have a doubling time of seconds. So despite departing from the hyperbolic model, we're back to predicting a singularity.
And I've seen no parsimonious theory for episodic models, by which I mean one or more differential equations whose solutions yield episodic growth. Differential equations are important for expressing how the state of a system affects changes in that state.
Something I'm interested in now is how to rectify that within a stochastic framework. Is there an elegant way to simulate episodic, stochastic acceleration in technological progress?
My own view of growth prospects is at this point black swan-style (even if the popularizer of that term called me a "BSer"). A stochastic hyperbolic model generates fat-tailed distributions for future growth and GWP, ones that imply that the expected value of future output is infinite. Leavening a conventional, insider prediction of stagnation with even a tiny bit of that outside view suffices to fatten its tails, send its expectation to infinity, and, as a practical matter, raise the perceived odds of extreme outcomes.
Thank you for this thoughtful response — and for all of your comments on the document! I agree with much of what you say here.
(No need to respond to the below thoughts, since they somehow ended up quite a bit longer than I intended.)
This is well put. I do agree with this point, and don’t want to downplay the value of taking outside view perspectives.
As I see it, there are a couple of different reasons to fit hyperbolic growth models — or, rather, models of form (dY/dt)/Y = aY^b + c — to historical growth data.
First, we might be trying to test a particular theory about the causes of the Industrial Revolution (Kremer’s “Two Heads” theory, which implies that pre-industrial growth ought to have followed a hyperbolic trajectory). Second, rather than directly probing questions about the causes of growth, we can use the fitted models to explore outside view predictions — by seeing what the fitted models imply when extrapolated forward.
I read Kremer’s paper as mostly being about testing his growth theory, whereas I read the empirical section of your paper as mostly being about outside-view extrapolation. I’m interested in both, but probably more directly interested in probing Kremer’s growth theory.
I think that different aims lead to different emphases. For example: For the purposes of testing Kremer’s theory, the pre-industrial (or perhaps even pre-1500) data is nearly all that matters. We know that the growth rate has increased in the past few hundred years, but that’s the thing various theories are trying to explain. What distinguishes Kremer’s theory from the other main theories — which typically suggest that the IR represented a kind of ‘phase transition’ — is that Kremer’s predicts an upward trend in the growth rate throughout the pre-modern era. So I think that’s the place to look.
On the other hand, if the purpose of model fitting is trend extrapolation, then there’s no particular reason to fit the model only to the pre-modern datapoint; this would mean pointlessly throwing out valuable information.
A lot of the reason I’m skeptical of Kremer’s model is that it doesn’t seem to fit very well with the accounts of economic historians and their descriptions of growth dynamics. His model seems to leave out too much and to treat the growth process as too homogenous across time. “Growth was faster in 1950AD than in 10,000BC mainly because there were more total ideas for new technologies each year, mainly because there were more people alive” seems really insufficient as an explanation; it seems suspicious that the model leaves out all of the other salient differences that typically draw economic historians’ attention. Are changes in institutions, culture, modes of production, and energetic constraints really all secondary enough to be slipped into the error term?
But one definitely doesn’t need to ‘believe’ the Kremer model — which offers one explanation for why long-run growth would follow a consistent hyperbolic trajectory — to find it useful to make growth extrapolations using simple hyperbolic models. The best case for giving significant weight to the outside view extrapolations, as I understand it, is something like (non-quote):
I do think this line of thinking makes sense, but in practice don’t update that much. While I don’t believe any very specific ‘inside view’ story about long-run growth, I do find it easy to imagine that was a phase change of one sort or another around the Industrial Revolution (as most economic historians seem to believe). The economy has also changed enough over the past ten thousand years to make it intuitively surprising to me that any simple unified model — without phase changes or piecewise components — could actually do a good job of capturing growth dynamics across the full period.
I think that a more general prior might also be doing some work for me here. If there’s some variable whose growth rate has recently increased substantially, then a hyperbolic model — (dY/dt)/Y = a*Y^b, with b > 0 — will often be the simplest model that offers an acceptable fit. But I’m suspicious that extrapolating out the hyperbolic model will typically give you good predictions. It will more often turn out to be the case that there was just a kind of phase change.
I think this is a completely valid criticism.
I agree that B > 0 is the more important hypothesis to focus on (and it’s of course what you focus on in your report). I started out investigating B = 1, then updated parts of the document to be about B > 0, but didn’t ultimately fully switch it over. Part of the issue is that B = 0 and B = 1 are distinct enough to support at least weak/speculative inferences from the radiocarbon graphs. This led me to mostly focus on B > 0 when talking about the McEvedy data, but focus on B = 1 when talking about the radiocarbon data. I think, though, that this mixing-and-matching has resulted in the document being somewhat confusing and potentially misleading in places.
I think that this is also a valid criticism: I never really say outright what would count as confirmation, in my mind.
Supposing we had perfectly accurate data, I would say that a necessary condition for considering the data “consistent” with the hypothesis is something like: “If we fit a model of form (dP/dt)/P = a*P^b to population data from 5000BC to 1700AD, and use a noise term that models stochasticity in a plausible way, then the estimated value of b should not be significantly less than .5”
I only ran this regression using normal noise terms, rather than using the more theoretically well-grounded approach you’ve developed, so it’s possible the result would come out different if I reran it. But my concerns about data quality have also had a big influence on my sloppiness tolerance here: if a statistical result concerning (specifically) the pre-modern subset of the data is sufficiently sensitive to model specification, and isn’t showing up in bright neon letters, then I’m not inclined to give it much weight.
(These regression results ultimately don’t have a substantial impact on my views, in either direction.)
I think this was an unclear statement on my part. I’m referring to the linear and non-linear regressions that Kremer runs on his population dataset (Tables II and IV), showing that population is significantly predictive of population growth rates for subsets that contain the Industrial Revolution. I didn’t mean to include his tests for heteroskedasticity or stability in that comment.
Just wanted to say that I believe this is useful too! Beyond the reasons you list here, I think that your modeling work also gives a really interesting insight into — and raises really interesting questions about — the potential for path-dependency in the human trajectory. I found it very surprising, for example, that re-rolling-out the fitted model from 10,000BC could give such a wide range of potential dates for the growth takeoff.
I think that it should make a difference, although you’re right to suggest that the difference may not be huge. If we were fully convinced that the episodic model was right, then one natural outside view perspective would be: “OK, the growth rate has jumped up twice over the course of human history. What the odds it will happen at least once more?”
This particular outside view should spit out a greater than 50% probability, depending on the prior used. It will be lower than the probability that hyperbolic trend extrapolation outside view spits out, but, by any conventional standard, it certainly won’t be low!
Whichever view of economic history we prefer, we should make sure to have our seatbelts buckled.
I’m saying Kremer’s “theory” rather than Kremer’s “model” to avoiding ambiguity: when I mention “models” in this comment I always mean statistical models, rather than growth models. ↩︎
I don’t know, of course, if Kremer would actually frame the empirical part of the paper quite this way. But if all the paper showed is that growth increased around the Industrial Revolution, this wouldn’t really be a very new/informative result. The fact that he’s also saying something about pre-modern growth dynamics (potentially back to 1 million BC) seems like the special thing about the paper — and the thing the paper emphasizes throughout. ↩︎
To stretch his growth theory in an unfair way: If there’s a slight low-hanging fruit effect, then the general theory suggests that — if you kept the world exactly as it was in 10000BC, but bumped its population up to 2020AD levels (potentially by increasing the size of the Earth) — then these hunter-gatherer societies would soon start to experience much higher rates of economic growth/innovation than what we’re experiencing today. ↩︎
I agree with much of this. A few responses.
I think the distinction between testing a theory and testing a mathematical model makes sense, but the two are intertwined. A theory will tend naturally to to imply a mathematical model, but perhaps less so the other way around. So I would say Kremer is testing both a theory and and model—not confined to just one side of that dichotomy. Whereas as far as I can see the sum-of-exponentials model is, while intuitive, not so theoretically grounded. Taken literally, it says the seeds of every economic revolution that has occurred and will occur were present 12,000 years ago (or in Hanson (2000), 2 million years ago), and it's just taking them a while to become measurable. I see no framework behind it that predicts how the system will evolve as a function of its current state rather than as a function of time. Ideally, the second would emerge from the first.
Note that what you call Kremer's "Two Heads" model predates him. It's in the endogenous growth theory of Romer (1986, 1990), which is an essential foundation for Kremer. And Romer is very much focused on the modern era, so it's not clear to me that "For the purposes of testing Kremer’s theory, the pre-industrial (or perhaps even pre-1500) data is nearly all that matters." Kuznets (1957) wrote about the contribution of "geniuses"—more people, more geniuses, faster progress. Julian Simon built on that idea in books and articles.
Actually, I believe the standard understanding of "technology" in economics includes institutions, culture, etc.—whatever affects how much output a society wrings from a given amount of inputs. So all of those are by default in Kremer's symbol for technology, A. And a lot of those things plausibly could improve faster, in the narrow sense of increasing productivity, if there are more people, if more people also means more societies (accidentally) experimenting with different arrangements and then setting examples for others; or if such institutional innovations are prodded along by innovations in technology in the narrower sense, such as the printing press.
Just on this point:
For the general Kremer model, where the idea production function is dA/dt = a(P^b)(A^c), higher levels of technology do support faster technological progress if c > 0. So you're right to note that, for Kremer's chosen parameter values, the higher level of technology in the present day is part of the story for why growth is faster today.
Although it's not an essential part of the story: If c = 0, then the growth is still hyperbolic, with the growth rate being proportional to P^(2/3) during the Malthusian period. I suppose I'm also skeptical that at least institutional and cultural change are well-modeled as resulting from the accumulation of new ideas: beneath the randomness, the forces shaping them typically strike me as much more structural.
Robert Gordon has argued for a coming growth slowdown: paper, book.
One item of feedback: I'd find the summary more satisfying if it gave a bit more detail on the analytic methods used to reach the conclusions. Basically, I understand the summary to say that the early data is noisy, and the new data doesn't fit a hyperbola. But does the data look hyperbolic despite the noise? What shape is the new data? Is there a systematic approach to fitting different models? What model classes are used? How is goodness of fit compared? Even a little such information could go a long way in helping readers to decide what to think, and whether to read the report in full.
Thanks for the feedback! I probably ought to have said more in the summary.
For the 'old data': I run a non-linear regression on the population growth rate as a function of population, for a dataset starting in 10000BC. The function is (dP/dt)/P = a*P^b, where P represents population. If b = 0, this corresponds to exponential growth. If b = 1, this corresponds to the strict version of the Hyperbolic Growth Hypothesis. If 0 < b < 1, this still corresponds to hyperbolic growth, although the growth rate is less than proportional to the population level. I found that if you start in 10000BC and keep adding datapoints, b is not significantly greater than 0 until roughly 1750 (although it is significantly less than 1). Here's a graph of how the value evolves.
Since the datapoints are unevenly spaced, it can make sense to weigh them in proportion to the length of the interval used to estimate the growth rate for that datapoint. If you do this, then b is actually significantly greater than 0 (although is still less than 1) for most of the interval. However, this is mostly driven by a single datapoint for the period from 10,000BC to 5,000BC. If you remove this single datapoint, which roughly corresponds to the initial transition to agriculture, then b again isn't significantly greater than 0 until roughly the Industrial Revolution. (Here are the equivalent graphs, with and without the initial datapoint.)
A key point is that, if you fit this kind of function to a dataset that includes a large stable increase in the growth rate, you'll typically find that b > 0. (For example: If you run a regression on a dataset where there's no growth before 1700AD, but steady 2% growth after 17000AD, you'll find that b is significantly greater than zero.) Mainly, it's a test of whether there's been a stable increase in the growth rate. So running the test on the full dataset (including the period around the IR) doesn't help us much to distinguish the hyperbolic growth story from the 'phase change'/'inflection point' story. Kremer's paper mainly emphasizes the fact that b approximately equals 1, when you run the regression on the full dataset; I think too much significance has sometimes been attributed to this finding.
If you just do direct curve fitting to the data -- comparing an exponential function and a hyperbolic function for b = 1 -- the exponential function is also a better fit for the period from 5000BC until the couple centuries before the Industrial Revolution. Both functions are roughly similarly bad if you throw in the 10,000BC datapoint. This comparison is just based on the mean squared errors of the two fits.
But I also think this data is really unreliable -- I'd classify a lot of the data points as something close to 'armchair guesses' -- so I don't think we should infer much either way.
There are also more recent datasets for particular regions (e.g. China) that estimate historical population growth curves on the basis of the relative number archeological deposits (such as human remains and charcoal) that have been dated to different time periods. There are various corrections that people do to try to account for things like the tendency of deposits to disappear or be destroyed over time. I found that it was a pain to recreate these population curves, from the available datasets, so I actually didn't do any proper statistical analysis using them. (Alex Lintz is currently doing this.)
Planned summary for the Alignment Newsletter:
Nit: Maybe you mean something stronger than transformative AI here? I don't know if it makes sense to me that future explosive growth should tell us much about timelines for transformative AI as traditionally defined (as a transition comparable to the agricultural or industrial revolution). If we know that neither past transition caused explosive growth, it feels like we should think it's quite plausible that transformative AI will have only a moderate impact on the growth rate.
On my read of this doc, everyone agrees that the industrial revolution led to explosive growth, and the question is primarily about whether we should interpret this as a one-off event, or as something that is likely to happen again in the future, so for all viewpoints it seems like transformative AI would still require explosive growth. Does that seem right to you?
My impression is that everyone agrees that the Industrial Revolution led to an increase in the growth rate, but that the Hyperbolic Growth Hypothesis (HGH) disagrees with the Series Of Exponentials Hypothesis on whether that increase in the growth rate was trend-breaking.
Put differently, the HGH says that as far as the growth rate is concerned, the Industrial Revolution wasn't special - or if it was, then it must attribute this to noise. According to the HGH, the growth rate has been increasing all the time according to the same hyperbolic function, and the industrial revolution was just a part of this trend. I.e. only one "growth mode" for all of history, rather than the industrial revolution ushering in a new growth mode.
By contrast, on the Series Of Exponential view, the Industrial Revolution did break the previous trend - we had exponential growth both before and after, but with different doubling times.
I agree with this, but it seems irrelevant to Asya's point? If it turned out to be the case that we would just resume the trend of accelerating growth, and AI was the cause of that, I would still call that transformative AI and I would still be worried about AI risks, to about the same degree as I would if that same acceleration was instead trend-breaking.
Yes, sorry, I think I was too quick to make a comment and should have paid more attention to the context. I think the claims in my comment are correct, but as you say it's not clear what exactly they're responding to, and in particular I agree it's not relevant to Asya's point on whether to use 'explosive'.
I think everyone agrees that the industrial revolution led to an increase in the growth rate. I think 'explosive' growth as Roodman talks about it hasn't happened yet, so I would avoid that term.
Ah, fair point, I'll change "explosive" to "accelerating" everywhere.
I think this is a misunderstanding. The common view is that the growth rate has been constant in the modern era.
The growth rate of output per person definitely has been roughly constant in developed countries (esp. the US) in the 20th century. In the doc, I'm instead talking about the growth rate of total output, globally, from about 1600 to 1950.
(So the summary may give the wrong impression. I ought to have suggested a tweak to make it clearer.)
Right, growth(GDP) > growth(GDP per capita) when growth(population)>0.
Although, is it the case that growth(GDP) increased during the modern era (ie, growth(population) has been rising)? My recollection is that the IR was a structural break, with g jumping from 0.5% to 2% (or something).
The world GDP growth rate also seems to have been increasing during the immediate lead-up to the Industrial Revolution, as well as during the following century, although the exact numbers are extremely uncertain. The growth rate most likely stabilized around the middle of the 20th century.
Do you know how mainstream the Hyperbolic Growth Hypothesis is? I was under the impression that the popular theory is that standards of living before the Industrial Revolution were Malthusian, with GDP chronically hovering near subsistence levels. But this is not my field of expertise.
I should have been clearer in the summary: the hypothesis refers to the growth rate of total economic output (GDP) rather than output-per-person (GDP per capita). Output-per-person is typically thought to have been pretty stagnant until roughly the Industrial Revolution, although just how stagnant it was is controversial. Total output definitely did grow substantially, though.
What l'm calling the Hyperbolic Growth Hypothesis is at least pretty mainstream. Michael Kremer's paper is pretty classic (it's been cited about 2000 times) and some growth theory textbooks repeat its main claim. Although I don't have a great sense of exactly how widely accepted it is.
There seems to be a major disconnect between the Hyperbolic Growth Hypothesis and the great divergence literature. If we take the Hyperbolic Growth Hypothesis seriously, it seems that there is really little to explain about the industrial revolution. It is just an inevitable consequence of hyperbolic growth and is not qualitatively distinct from what occured before. Although I'm not an economic historian, I have read a number of books on the great divergence and none of them seem to agree with that analysis. They may be disagreement about the causes and the precise timeline, but not about the existence of a question to be answered.
As to why they believe this, I think it essentially boils down to the fact that if we look at the historical record, it seems that the industrial revolution occuring c1800 was highly contingent. It seems unlikely that an observer in 500AD, even with excellent data about the past and detailed knowledge of future possible technologies, could have simply extrapolated growth trends to predict the industrial revolution. We know of a number of economically advanced societies which didn't industrialize, and indeed didn't appear to be on the path to industrialization. Examples include early modern China, Japan, or the Ottoman empire, or more tenously Song China or Early Imperial Rome. If northwestern Europe was more like China in the year 1600, industrialization may have taken much longer, even though in that conterfactual universe economic growth up to that point may have been similar to our own universe. Ditto for a conterfactual where the Americas didn't exist. So it seems that the Hyperbolic Growth Hypothesis proves too much, and isn't compatible with what we actually know about the industrial revolution.
I also pretty strongly have this intuition: the Kremer model, and the explanation it gives for the Industrial Revolution, is in tension with the impressions I've formed from reading the great divergence literature.
Although, to echo Max's comment, you can 'believe' the Kremer model without also thinking that an 18th/19th century Industrial Revolution was inevitable. It depends on how much noise you allow.
One of the main contributions in David Roodman's recent report is to improve our understanding of how noise/stochasticity can result in pretty different-looking growth trajectories, if you roll out the same hyperbolic growth model multiple times. For example, he fits a stochastic model to data from 10000BC to the present, then reruns the model using the fitted parameters. In something like a quarter of the cases, the model spits out a growth takeoff before 1AD.
I believe the implied confidence interval, for when the Industrial Revolution will happen, gets smaller and smaller as you move forward through history. I'm actually not sure, then, how inevitable the model says the IR would be by (e.g.) 1000AD. If it suggests a high level of inevitability in the timing, for instance implying the IR basically had to happen by 2000, then that would be cause for suspicion; the model would likely be substantially understating contingency.
(As one particular contingency you mention: It seems super plausible to me, especially, that if the Americas didn't turn out to exist, then the Industrial Revolution would have happened much later. But this seems like a pretty random/out-of-model fact about the world.)
I think Roodman's model implies a standard deviation of around 500-1000 years for IR timing starting from 1000AD, but I haven't checked. In general for models of this type it seems like the expected time to singularity is a small multiple of the current doubling time, with noise also being on the order of the doubling time.
The model clearly underestimates correlations and hence the variance here---regardless of whether we go in for "2 revolutions" or "randomly spread out" we can all agree that a stagnant doubling is more likely to be followed by another stagnant doubling and vice versa, but the model treats them as independent.
This seems to suggest there are lots of civilizations like Europe-in-1700. But it seems to me that by this time (and so I believe before the Americas had any real effect) Europe's state of technological development was already pretty unprecedented. This is lot of what makes many of the claims about "here's why the IR happened" seem dubious to me.
My sense of that comes from: (i) in growth numbers people usually cite, Europe's growth was absurdly fast from 1000AD - 1700AD (though you may think those numbers are wrong enough to bring growth back to a normal level) (ii) it seems like Europe was technologically quite far ahead of other IR competitors.
I'm curious about your take. Is it that:
I'm curious what numbers you are using for Europe's growth between 1000-1700; I didn't think European growth over that period was particularly unusual. It is worth remembering that Europe in 1000 (particularly northern Europe) was a backwater and so benefitted from catchup growth relative to (say) China. I also don't know how much of European growth was driven by extensive growth in eastern Europe, which doesn't seem to be relevant that to the great divergence.
Arguments against the idea that Europe c1700 was technologically ahead of the rest of Eurasia (r at least, China) are common in the great divergence literature. A good recent discussion is Chapter 16 of A Culture of Growth by Mokyr; he discusses various similarities and differences between the two regions around 1700. For detailed discussion focussed on military questions, see The Gunpowder Age by Andrade and Why did Europe conquer the world? by Hoffman, both of which argue that the gap between European and Chinese military technology was not very large during the 1600s.
For what it's worth I think Europe development was distinct from previous economic efflorescences in so far as it took place in the context of a fractured political landscape. Most other examples (Rome, Abbasid caliphate, many Chinese dynasties) seem to be driven by political unification allowing the growth and diversification of markets; a discussion focussed on the roman example can be found in The Roman Market Economy by Temin. This seems different to the situation in Europe c1700.
For what it's worth it seems to me that the most plausible explanations for the great divergence are rooted in European fragmentation. This allowed a number of different economic, political, and cultural arrangements to be explored while competitive pressure encouraged more efficient institutions to be adopted. A recent discussion of this can be found in Escape from Rome by Schiedel, but the argument is made in many other places and underpins a number of other more proximate explanations of the great divergence (including Mokyr's cultural/institutional explanation).
I took numbers from Wikipedia but have seen different numbers that seem to tell the same story although their quantitative estimates disagree a ton.
The first two numbers are all higher than growth rates could have plausibly been in a sustained way during any previous part of history (and the 0-1000AD one probably is as well), and they seem to be accelerating rather than returning to a lower mean (as must have happened during any historical period of similar growth).
My current view is that China was also historically unprecedented at that time and probably would have had an IR shortly after Europe. I totally agree that there is going to be some mechanistic explanation for why europe caught up with and then overtook china, but from the perspective of the kind of modeling we are discussing I feel super comfortable calling it noise (and expecting similar "random" fluctuations going forward that also have super messy contingent explanations).
[Caveat to all of the below is that these are vague impressions, based on scattered reading. I invite anyone with proper economic history knowledge to please correct me.]
I'm reasonably sympathetic to the first possibility. I think it’s somewhat contentious whether Europe or China was more ‘developed’ in 1700. In either case, though, my impression is that the state of Europe in 1700 was non-unprecedented along a number of dimensions.
The error bars are still pretty large here, but it’s common to estimate that Europe’s population increased by something like 50% between 1500 and 1700. (There was also probably a surge between something like 1000AD and 1300AD, as Western Europe sort of picked itself back up from a state of collapse, although I think the actual numbers are super unclear. Then the 14th century has famine and the Black Death, which Europe again needs to recover from.)
Something like a 50% increase over a couple centuries definitely couldn’t have been normal, but it’s also not clearly unprecedented. It seems like population levels in particular regions tended to evolve through a series of surges and contractions. We don't really know these numbers — although, I think, they’re at least inspired by historical records — but the McEvedy/Jones estimates show a 100% population increase in two centuries during the Song Dynasty (1000AD - 1200AD). We super don't know most of these numbers, but it seems conceivable that other few-century efflorescences were associated with similar overall growth rates: for example, the Abbasid Caliphate, the Roman Republic/Empire during its rise, the Han dynasty, the Mediterranean in the middle of the first century BCE.
These numbers are also presumably sketchy, but England’s estimated GDP-per-capita in 1700AD was also roughly the same as China’s estimated GDP-per-capita in 1000AD (according to a chart in British Economic Growth, 1270-1870); England is also thought to have been richer than other European states, with the exception of the Netherlands.
My impression is that Northwestern Europe’s growth from 1500 to 1700 also wasn’t super innovation-driven: a lot of it was about stuff like expanded trade networks and better internal markets. The maritime technology that supported global trade was enabled by innovation, but (I think) the technology wasn't obviously better than Chinese maritime technology in previous centuries. (E.g. Zheng He.) I think the technological progress that was happening at this point also wasn’t obviously more impressive than the sort of technological progress that happened in China in previous eras. Vaclav Smil (in Transforming the 20th Century) thinks the most technologically innovative time/place in history before 19th century Britain was early Han Dynasty China (roughly 200BC-1AD). The Song Dynasty (1000AD-1300AD) also often gets brought up. I don’t personally know a lot of details about the innovations produced during these periods, although I believe a number of them were basically early (and sometimes better) versions of later European innovations. One specific claim I've encountered is that the volume of iron/steel production was plausibly about the same in 1000AD Song China and in 1700AD Europe.
Here is one good/classic paper on previous economic efflorescences and their implications for our understanding of the Industrial Revolution. I also pulled out a few different long quotes, to make a 3 page summary version here.
If one believed the numbers on wikipedia, it seems like Chinese growth was also accelerating a ton and it was not really far behind on the IR, such that I wouldn't except to be able to easily eyeball the differences.
If you are trying to model things at the level that Roodman or I are, the difference between 1400 and 1600 just isn't a big deal, the noise terms are on the order of 500 years at that point.
So maybe the interesting question is if and why scholars think that China wouldn't have had an IR shortly after Europe (i.e. within a few centuries, a gap small enough that it feels like you'd have to have an incredibly precise model to be justifiably super surprised).
Maybe particularly relevant: is the claimed population growth from 1700-1800 just catch-up growth to Europe? (more than doubling in 100 years! And over the surrounding time period the observed growth seems very rapid even if there are moderate errors in the numbers) If it is, how does that work given claims that Europe wasn't so far ahead by 1700? If it isn't, then how does the that not very strongly suggest incredible acceleration in China, given that it had very recently had some of the fastest growth in history and is then experience even more unprecedented growth? Is it a sequence of measurement problems that just happen to suggest acceleration?
I believe the population surge is closely related to the European population surge: it's largely attributed to the Colombian exchange + expanded markets/trade. One of the biggest things is that there's an expansion in the land under cultivation, since potatoes and maize can be grown on marginal land that wouldn't otherwise work well for rice or wheat, and (probably) a decline in living standards that's offsetting the rise in population. From the book 1493 (ch. 5):
There's obviously a major risk of hindsight bias here, but I think there's almost a consensus among economic historians that China wasn't on track toward an industrial revolution anytime soon. There aren't really signs of innovation picking up during this period: "the prosperity engendered by quantitative growth in output masked the lack of significant innovation in productive technologies" (The Economic History of China, p. 336). Estimates seem to vary widely, and I don't know what the error bars are here, but the favored estimates in TECHC (taken from a Chinese-language paper by Liu Ti) also show the industrial sector of the economy actually shrinking by half between 1600 and 1840 and real per-capita incomes shrinking by about a quarter.
It's also a common view that China was entering a period of decline at the start of the nineteenth century (partly due to population pressure and ecological damage from land conversion). From the same book (p. 361):
Basically, I think the story is that: There was another 2-3 century "efflorescence" in China, but it wasn't really associated with either technological innovation or an expansion of industry. The total population growth numbers were probably unusually big, relative to other efflorescences, but this doesn't imply that this was an unusually innovative period; the unusual size of the surge may just reflect the fact that there was a black-swan-ish ecological event (the sudden transfer of several New World crops) around the start of the period. The growth surge was unsustainable, as all previous growth surges had been, and China was on track to fall back down to a lower level of development.
EDIT: One more quote, from A Culture of Growth (p. 317; emph. mine):
Thanks, super helpful.
(I don't really buy an overall take like "It seems unlikely" but it doesn't feel that mysterious to me where the difference in take comes from. From the super zoomed out perspective 1200 AD is just yesterday from 1700AD, it seems like random fluctuations over 500 years are super normal and so my money would still be on "in 500 years there's a good chance that China would have again been innovating and growing rapidly, and if not then in another 500 years it's reasonably likely..." It makes sense to describe that situation as "nowhere close to IR" though. And it does sound like the super fast growth is a blip.)
I agree this is puzzling, and I'd love to see more discussion of this.
However, it seems to be that at least in principle there could be a pretty boring explanation: The HGH is correct about the fundamental trend, and the literature on the Industrial Revolution has correctly identified (and maybe explained) a major instance of noise.
Note also that the phenomenon that social behavior that is individually contingent is nevertheless governed by simple macro-laws with few parameters is relatively ubiquitous. E.g. the exact timing of all major innovations since the Industrial Revolution (electricity, chemical engineering, computers, ...) seems fairly contingent, and yet overall the growth rate is remarkably close to constant. Similarly for the rest of Kaldor's facts.