Ben Garfinkel

Ben Garfinkel - Researcher at Future of Humanity Institute

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Does Economic History Point Toward a Singularity?

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.

Does Economic History Point Toward a Singularity?

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.)

Does Economic History Point Toward a Singularity?

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.


  1. 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. ↩︎

  2. 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. ↩︎

Does Economic History Point Toward a Singularity?

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 input. 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.

Does Economic History Point Toward a Singularity?

Hi David,

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.)

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 foreseight 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.

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).[1] 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.[2] 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?[3]

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):

We know that growth rates permanently increased in the centuries around the Industrial Revolution. Constant exponential growth models therefore fit long-run growth data terribly. Models of form (dY/dt)/Y = aY^b can fit the data much better, since they allow the growth rate to increase. If we fit one of these models to the long-run growth data (with an error term to account for stochasticity) we find that b > 0, implying hyperbolically increasing growth rates. Extrapolated forward, this implies that infinite rates are nearly inevitable in the future. While we we of course know that growth won’t actually become infinite, we should still update in the direction of believing that much faster growth is coming, because this is the simplest model that offers an acceptably good fit, and because we shouldn’t be too confident in any particular inside view model of how economic growth works.

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.

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 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.

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.

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 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."

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.

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 infinitesimal moment feed into the growth process to shape the probability distribution for growth over finite periods such as 10 years. This seemed potentially useful for two reasons. For one, since it was hard to do, 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.

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.

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.

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.


  1. 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. ↩︎

  2. 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. ↩︎

  3. 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. ↩︎

Asking for advice

I would also like to come out of the woodwork as someone who finds Calendly vaguely annoying, for reasons that are entirely opaque to me.

(Although it's also unambiguously more convenient for me when people send me Calendly links -- and, given the choice, I think I'd mostly like people to keep doing this.)

Does Economic History Point Toward a Singularity?

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.

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):

Neither rice nor wheat, China’s two most important staples, would grow in the shack people’s marginal land. The soil was too thin for wheat; on steep slopes, the irrigation for rice paddies requires building terraces, the sort of costly, hugely laborious capital improvement project unlikely to be undertaken by renters. Almost inevitably, they turned to American crops: maize, sweet potato, and tobacco. Maize (Zea mays) can thrive in amazingly bad land and grows quickly, maturing in less time than barley, wheat, and millet. Brought in from the Portuguese at Macao, it was known as “tribute wheat,” “wrapped grain,” and “jade rice.” Sweet potatoes will grow where even maize cannot, tolerating strongly acid soils with little organic matter and few nutrients....

In their quest for social stability, the Ming had prohibited people from leaving their home regions. Reversing course, the Qing actively promoted a westward movement. Much as the United States encouraged its citizens to move west in the nineteenth century and Brazil provided incentives to occupy the Amazon in the twentieth, China’s new Qing masters believed that filling up empty spaces was essential to the national destiny.... Lured by tax subsidies and cheap land, migrants from the east swarmed into the western hills.... They looked at the weathered, craggy landscape, so unwelcoming to rice—and they, too, planted American crops....

The amount of cropland soared, followed by the amount of food grown on that cropland, and then the population.

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):

[T]he economic growth of the nineteenth century could not be sustained indefinitely. There is considerable evidence that the Chinese economy had seriously begun to exhaust its productive capacities by 1800.

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):

We will never know whether without the rise of the West, the Orient would have been able to replicate something similar, given enough time. It seems unlikely, but there is no way of knowing if they would have stumbled upon steam power or the germ theory of disease. It is true that the consensus of modern scholarship has remained of the opinion that by 1800 the bulk of output in Chinese industry employed a technology very little different from that under the Song (Richardson, 1999, pp. 54–55). At the level of the economy as a whole, this is an overstatement: Chinese agriculture adopted new crops such as peanuts and sweet potatoes, some of which were introduced by the intercontinental ecological arbitrage practiced by European explorers in the sixteenth century. Stagnation is therefore too strong a word, but comparing Chinese technological achievements not only with those of the West but also with its own successes during the Song clearly indicates a decelerating progress. Elvin (1996, p. 93), after studying the missed opportunities of hydraulic technology adoption in China, concludes that there were strong and perceived needs, and few constraints in adopting such techniques. And yet there was minimal advance. China’s technological somnolence was rudely interrupted by the exposure to Western technology in the nineteenth century.

Does Economic History Point Toward a Singularity?

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:

  • The world wasn't yet historically exceptional by 1700, there have been other comparable periods of rapid progress. (What are the historical analogies and how analogous do you think they are? Is my impression of technological sophistication wrong?)

  • 1700s Europe is quantitatively exceptional by virtue of being the furthest along example, but nevertheless there is a mystery to be explained about why it became even more exceptional rather than regressing to the mean (as historical exceptional-for-their-times civilizations had in the past). I don't currently see a mystery about this (given the level of noise in Roodman's model, which seems like it's going to be in the same ballpark as other reasonable models), but it may be because I'm not informed enough about those historical analogies.

  • Actually the IR may have been inevitable in 1700s Europe but the exact pace seems contingent. (This doesn't seem like a real tension with a continuous acceleration model.)

  • Actually the contingencies you have in mind were already driving the exceptional situation in 1700.

[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.

Does Economic History Point Toward a Singularity?

Thanks for the feedback! I probably ought to have said more in the summary.

Essentially:

  • 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.)

    • I went entirely off of the graphs and summary statistics given in papers analyzing these datasets, which tend to be interested in pretty different questions. In short: Most of the graphs show pretty huge and condensed growth spikes, which the authors often attribute to the beginning of intensive agriculture within the region; in many of the graphs, the spikes are followed by roughly flat or even declining population levels. The implied population growth rates for the few-thousand-year-periods containing the spikes are also typically comparable to the (admittedly unreliable) population growth rates that people have estimated for the period from 1AD to 1500AD.
Does Economic History Point Toward a Singularity?

Economic histories do tend to draw casual arrows between several of these differences, sometimes suggesting a sort of chain reaction, although these narrative causal diagrams are admittedly never all that satisfying; there’s still something mysterious here.

Just to make this more concrete:

One example of an IR narrative that links a few of these changes together is Robert Allen's. To the extent that I understand/remember it, the narrative is roughly: The early modern expansion of trade networks caused an economic boom in England, especially in textile manufacturing. As a result, wages in England became unusually high. These high wages created unusually strong incentives to produce labor-saving technology. (One important effect of the Malthusian conditions is that they make labor dirt cheap.) England, compared to a few other countries that had similarly high wages at other points in history, also had access to really unusually cheap energy; they had huge and accessible coal reserves, which they were already burning as a replacement for wood. The unusually high levels of employment in manufacturing and trade also supported higher levels of literacy and numeracy. These conditions came together to support the development of technologies for harnessing fossil fuels, in the 19th century, and the rise of intensive R&D; these may never have been economically rational before. At this point, there was now a virtuous cycle that allowed England's growth -- which was initially an unsustainable form of growth based on trade, rather than technological innovation -- to become both sustained and innovation-driven. The spark then spread to other countries.

This particular tipping point story is mostly a story about why growth rates increased from the 19th century onward, although the growth surge in the previous few centuries, largely caused by the Colombian exchange and expansion of trade networks, still plays an important causal role; the rapid expansion of trade networks drives British wages up and makes it possible for them to profitably employ a large portion of their population in manufacturing.

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