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.
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.