This would be an important update for me, so I'm excited to see people looking into it and to spend more time thinking about it myself.

High-level summary of my current take on your document:

• I agree that the 1AD-1500AD population data seems super noisy.
• Removing that data removes one of the datapoints supporting continuous acceleration (the acceleration between 10kBC - 1AD and 1AD-1500AD) and should make us more uncertain in general.
• It doesn't have much net effect on my attitude towards continuous acceleration vs discontinuous jumps, this mostly pushes us back towards our prior.
• I'm not very moved by the other evidence/arguments in your doc.

Here's how I would summarize the evidence in your document:

• Much historical data is made up (often informed by the author's models of population dynamics), so we can't use it to estimate historical growth. This seems like the key point.
• In particular, although standard estimates of growth from 1AD to 1500AD are significantly faster than growth between 10kBC and 1AD, those estimates are sensitive to factor-of-1.5 error in estimates of 1AD population, and real errors could easily be much larger than that.
• Population levels are very noisy (in addition to population measurement being noisy) making it even harder to estimate rates.
• Radiographic data often displays isolated periods of rapid growth from 10,000BC to 1AD and it's possible that average growth rates were something like 2000 year doubling. So even if 500-2000 year doubling times are accurate from 1AD to 1500, those may not be a deviation from the preceding period.
• You haven't looked into the claims people have made about growth from 100kya to 10kya, but given what we know about measurement error from 10kya to now, it seems like the 100kya-10kya data is likely to be way too noisy to say anything about.

Here's my take in more detail:

• You are basically comparing "Series of 3 exponentials" to a hyperbolic growth model. I think our default simple hyperbolic growth model should be the one in David Roodman's report (blog post), so I'm going to think about this argument as comparing Roodman's model to a series of 3 noisy exponentials. In your doc you often dunk on an extremely low-noise version of hyperbolic growth but I'm mostly ignoring that because I absolutely agree that population dynamics are very noisy.
• It feels like you think 3 exponentials is the higher prior model. But this model has many more parameters to fit the data, and even ignoring that "X changes in 2 discontinuous jumps" doesn't seem like it has a higher prior than "X goes up continuously but stochastically." I think the only reason we are taking 3 exponentials seriously is because of the same kind of guesswork you are dismissive of, namely that people have a folk sense that the industrial revolution and agricultural revolutions were discrete changes. If we think those folk senses are unreliable, I think that continuous acceleration has the better prior. And at the very least we need to be careful about using all the extra parameters in the 3-exponentials model, since a model with 2x more parameters should fit the data much better.
• On top of that, the post-1500 data is fit terribly by the "3 exponentials" model. Given that continuous acceleration very clearly applies in the only regime where we have data you consider reliable, and given that it already seemed simpler and more motivated, it seems pretty clear to me that it should have the higher prior, and the only reason to doubt that is because of growth folklore. You can't have it both ways in using growth folklore to promote this hypothesis to attention and then dismissing the evidence from growth folklore because it's folklore.
• On the acceleration model, the periods from 1500-2000, 10kBC-1500, and "the beginning of history to 10kBC" are roughly equally important data (and if that hypothesis has higher prior I don't think you can reject that framing). Changes within 10kBC - 1500 are maybe 1/6th of the evidence, and 1/3 of the relevant evidence for comparing "continuous acceleration" to "3 exponentials." I still think it's great to dig into one of these periods, but I don't think it's misleading to present this period as only 1/3 of the data on a graph.
• (Enough about priors, onto the data.)
• I think that the key claim is that the 1AD-1500AD data is mostly unreliable. Without this data, we have very little information about acceleration from 10kBC - 1500AD, since the main thing we actually knew was that 1AD-1500AD must have been faster than the preceding 10k years. I'd like to look into that more, but it looks super plausible to me that the noise is 2x or more for 1AD which is enough to totally kill any inference about growth rates. So provisionally I'm inclined to accept your view there.
• That basically removes 1 datapoint for the continuous acceleration story and I totally agree it should leave us more uncertain about what's going on. That said, throwing out all the numbers from that period also removes one of the main quantitative datapoints against continuous acceleration [ETA: the other big one being the modern "great stagnation," both of these are in the tails of the continuous acceleration story and are just in the middle of the constant exponentials in the 3-exponential story, though see Robin Hanson's writeup to get a sense for what the series of exponentials view actually ends up looking like---it's still surprised by the great stagnation], and comes much closer to leaving us with our priors + the obvious acceleration over longer periods + the obvious acceleration during the shorter period where we actually have data, which seem to all basically point in the same direction.
• Even taking the radiocarbon data as given I don't agree with the conclusions you are drawing from that data. It feels like in each case you are saying "a 2-exponential model fits fine" but the 2 exponentials are always different. The actual events (either technological developments or climate change or population dynamics) that are being pointed to as pivotal aren't the same across the different time series and so I think we should just be analyzing these without reference to those events (no suggestive dotted lines :) ). I spent some time doing this kind of curve fitting to various stochastic growth models and this basically looks to me like what individual realizations look like from such models--the extra parameters in "splice together two unrelated curves" let you get fine-looking fits even when we know that the underlying dynamics are continuous+stochastic.
• I currently don't trust the population data coming from the radiocarbon dating. My current expectation is that after a deep dive I would not end up trusting the radiocarbon dating at all for tracking changes in the rate of population growth when the populations in question are changing how they live and what kinds of artifacts they make (from my perspective, that's what happened with the genetics data, which wasn't caveated so aggressively in the initial draft I reviewed). I'd love to hear from someone who actually knows about these techniques or has done a deep dive on these papers though.
• I think the only dataset that you should expect to provide evidence on its own is the China population time series. But even there if you just take rolling averages and allow for a reasonable level of noise I think the continuous acceleration story looks fine. E.g. I think if you compare David Roodman's model with the piecewise exponential model (both augmented with measurement noise, and allowing you to choose noisy dynamics however you want for the exponential model), Roodman's model is going to fit the data better despite having fewer free parameters. If that's the case, I don't think this time series can be construed as evidence against that model.
• I agree with the point that if growth is 0 before the agricultural revolution, rather than "small," then that would undermine the continuous acceleration story. I think prior growth was probably slow but non-zero, and this document didn't really update my view on that question.

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:

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.

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

For most of human history, up until the twentieth century, the economic growth rate has tended to be roughly proportional to the level of economic output.

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.

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):

• Each individual technological or institutional boost to economic growth theoretically follows a logarithmic pattern. That is, a particular technology's adoption has diminishing marginal returns, and is ultimately finite. Classic hypothetical example is the tractor superseding the plough - the first few hundred adopters may see huge gains to productivity, followed by some for whom it's good but not amazing, followed by the part-time farmers let's say, then the hobbyists or something, until every farm uses the new technology. The same diminishing marginal returns pattern applies to every single event, or level effect, that raises GDP (or GDP/capita), regardless of the kind of growth, be it extensive, intensive, or Smithian - so lowering tariffs (finite, as you can't lower them below 0), conquest (you eventually run out of lands to conquer), particular technologies (you run out of lands to which to apply the new tractor).
• So when we look at the overall curves, what appears to be exponential is, at the more zoomed in level, really just a whole lot of logarithmic curves strung together. When those curves are closer together, then it can appear exponential, as it has for the past two or three centuries. When they are a further apart, there appears to be stagnation. So if we're lucky, then the rate at which we have things that give us level effects increases, such that we never approach the stage where things plateau.
• Tangentially, but importantly, the effect of each level effect on overall growth rates is very much determined by the size of the sector of the economy that they affect. So you might get an astonishing improvement to manufacturing in, say, the year 1300, but as the vast majority of the economy was agricultural, it might thus have a negligible effect on overall measured GDP. (A similar point is neatly illustrated in Gregory Clark's book Farewell to Alms, on p.255, fig.12.12, which compares the real purchasing power of workers with historical vs modern tastes - from the perspective of a farm labourer at the time, living standards according to his data in the Middle Ages were only superseded in the 19thC after a period of decline; when using the typical consumption basket of an American middle-class person today, however, there was steady improvement since about 1500).
• The same principles apply for service improvements in the era when the majority of GDP was in manufacturing. And the same today, in which there have lately been astonishing advancements in the productivity of agriculture and manufacturing - I mean truly mind-blowing stuff that would make Jethro Tull or Richard Arkwright break down in tears if they saw them - but because they are now such a small proportion of overall GDP (and employment, too) they don't appear to make much difference to the overall figures.

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.

Addendum:

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.^[1] 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):

The basic facts are not in dispute. The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society. Of course, innovation has taken place throughout history. Milestone breakthroughs in earlier times—such as water mills, the horse collar, and the printing press—can all be traced more or less, and their economic effects can be assessed. They appeared, often transformed an industry affected, but once incorporated, further progress slowed and sometimes stopped altogether. They did not trigger anything resembling sustained technological progress....

The early advances in the cotton industry, iron manufacturing, and steam power of the years after 1760 became in the nineteenth century a self-reinforcing cascade of innovation, one that is still very much with us today and seems to grow ever more pervasive and powerful. If economic growth before the Industrial Revolution, such as it was, was largely driven by trade, more effective markets and improved allocations of resources, growth in the modern era has been increasingly driven by the expansion of what was known in the age of Enlightenment as “useful knowledge” (A Culture of Growth, p. 3-4).

Technological modernity is created when the positive feedback from the two types of knowledge [practical knowledge and scientific knowledge] becomes self-reinforcing and autocatalytic. We could think of this as a phase transition in economic history, in which the old parameters no longer hold, and in which the system’s dynamics have been unalterably changed ("Long-Term Economic Growth and the History of Technology", p. 12).

And here’s Robert Allen telling another phase transition story (quotes stitched together from Global Economic History: A Very Short Introduction):

The greatest achievement of the Industrial Revolution was that the 18th-century inventions were not one-offs like the achievements of earlier centuries. Instead, the 18th-century inventions kicked off a continuing stream of innovations.

The explanation [for the Industrial Revolution] lies in Britain’s unique structure of wages and prices. Britain’s high-wage, cheap-energy economy made it profitable for British firms to invent and use the breakthrough technologies of the Industrial Revolution.

The Western countries have experienced a development trajectory in which higher wages led to the invention of labour-saving technology, whose use drove up labour productivity and wages with it. The cycle repeats.

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]

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):

This discussion therefore suggests that models based on economies of scale of one sort or another do not provide us with fundamental causes of cross-country income differences. At best, they are theories of growth of the world taken as a whole. Moreover, once we recognize that the modern economic growth process has been uneven, meaning that it took place in some parts of the world and not others, the appeal of such theories diminishes further. If economies of scale were responsible for modern economic growth, this phenomenon should also be able to explain when and where this process of economic growth started. Existing models based on economies of scale do not. In this sense, they are unlikely to provide the fundamental causes of modern economic growth.

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:

The [historical] record is that despite many significant, even pathbreaking innovations in many societies since the start of written history, [technological progress] has not really been a major factor in economic growth, such as it was, before the Industrial Revolution.... The Industrial Revolution, then, can be regarded not as the beginnings of growth altogether but as the time at which technology began to assume an ever-increasing weight in the generation of growth and when economic growth accelerated dramatically ("Long-Term Economic Growth and the History of Technology," pg. 1116-118).

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.

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.

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.

Planned summary for the Alignment Newsletter:

One important question for the long-term future is whether we can expect accelerating growth in the near future (see e.g. this <@recent report@>(@Modeling the Human Trajectory@)). For AI alignment in particular, the answer to this question could have a significant impact on AI timelines: if some arguments suggested that it would be very unlikely for us to have accelerating growth soon, we should probably be more skeptical that we will develop transformative AI soon.

So far, the case for accelerating growth relies on one main argument that the author calls the _Hyperbolic Growth Hypothesis_ (HGH). This hypothesis posits that the growth _rate_ rises in tandem with the population size (intuitively, a higher population means more ideas for technological progress which means higher growth rates). This document explores the _empirical_ support for this hypothesis.

I’ll skip the messy empirical details and jump straight to the conclusion: while the author agrees that growth rates have been increasing in the modern era (roughly, the Industrial Revolution and everything after), he does not see much support for the HGH prior to the modern era. The data seems very noisy and hard to interpret, and even when using this noisy data it seems that models with constant growth rates fit the pre-modern era better than hyperbolic models. Thus, we should be uncertain between the HGH and the hypothesis that the industrial revolution triggered a one-off transition to increasing growth rates that have now stabilized.

Planned opinion:

I’m glad to know that the empirical support for the HGH seems mostly limited to the modern era, and may be weakly disconfirmed by data from the pre-modern era. I’m not entirely sure how I should update -- it seems that both hypotheses would be consistent with future accelerating growth, though HGH predicts it more strongly. It also seems plausible to me that we should still assign more credence to HGH because of its theoretical support and relative simplicity -- it doesn’t seem like there is strong evidence suggesting that HGH is false, just that the empirical evidence for it is weaker than we might have thought. See also Paul Christiano’s response.

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.

Robert Gordon has argued for a coming growth slowdown: paper, book.