A Discussion of Other Models for Population Growth
[This is part of my series on Thomas Malthus’ “Essay on the Principle of Population,” first published in 1798. You can find an overview of all my posts here that I will keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?”]
The thrust of my posts is to propose a non-Malthusian explanation for population dynamics. This is independent from my critique of the Malthusian argument. If it turned out that my theory does not hold up, this would not do away with my critique or validate what it is directed against. If instead it turned out that my theory yields a better explanation for population dynamics, it strengthens the critique because it blocks an argument that the Malthusian theory is the best we have. To achieve this, my proposal need not be perfect, it only has to work considerably better.
Since I am certainly not the first one to think about all this, I have tried to look at as many theories for population dynamics as I could find. My search may have been incomplete, though. If you are aware of other approaches that I miss, I am very eager to hear about them. Maybe someone has made a similar proposal as mine already. Then I will cede my precedence. Or they had some other idea that I missed. Then I would want to see how it stacks up against my theory or whether it can be integrated. Please leave any suggestions in the comments. All hints are very much appreciated.
In the following, I will discuss the main theories for population dynamics that I am currently aware of: Logistic growth, hyperbolic growth, and an adaptation of the Malthusian model to a two-sector economy as proposed by Lemin Wu and also Hans-Joachim Voth and Nico Voigtländer. All three approaches do better than the Malthusian argument and add value. They get at important things about population dynamics in my view. However, I think the authors are still in thrall to the Malthusian argument and only try to fix it although it cannot be repaired. In addition, I will review also various ideas proposed by critics of Malthus, often going back to the 19th century, that are not really models, but may play a role in them.
In this post, I will focus on logistic and hyperbolic growth models. The rest will follow in one or maybe two further posts.
Logistic Growth
The idea to make the Malthusian model less abrupt leads to a very old proposal. As far as I can see, the Belgian mathematician Pierre François Verhulst (1804–1849) was the first to come up with it. But also others had the idea as it seems independently: the biologist Raymond Pearl (1879–1940), the statistician Lowell Reed (1886–1966), and the mathematician Alfred Lotka (1880–1949). However, that was much later.
The stark conclusion in Malthus’ original model for a population “when unchecked” is that it runs at maximum fertility no matter what. That implies exponential growth if mortality is also fixed. When you introduce a hard constraint for the food supply, which means a maximum size for the population that cannot be breached or at least not for long, it follows that the population bumps into it at full speed and comes to a halt only because of steeply rising mortality from starvation and famine.
Malthus himself tries to soften this a little by introducing also the “positive checks” that can raise mortality already below the maximum size possible without them. If you assume that mortality rises with population size, that means population growth begins to slow down somewhat earlier. Malthus was unable to grasp what that meant mathematically because he only knew about the linear and the exponential function. And not even that, he was unable to think in continuous time and had to hammer it out in the discrete setup with fixed absolute and fixed proportional increases from one generation to the next: the arithmetical and geometrical ratios.
Verhulst basically fills this gap in Malthus’ exposition. Instead of a fixed rate of growth, he makes the rate dependent on population size by introducing a modulating factor that falls linearly from 1 for a population size of 0 to 0 for a population of maximum size. This extra input tunes the growth rate down with increasing numbers. The result is that the population first starts out with fast growth (not exponential growth as many people claim), then reaches an inflection point at half the maximum size, and eventually decelerates and approaches the maximum from below with ever slower growth.
The functional form is known as the logistic function or also the “sigmoidal” function alluding to the “S” shape although the term is sometimes used more generally for similar functions. It looks like this (picture from Wikipedia):
If you are interested in spelledd-out formulas, you can find a discussion on Wikipedia. But I think it is entirely sufficient to get the intuitive idea: The function starts out with growth at a high rate, which then slows down to zero as the size of the population grows to the maximum possible.
Here is another way to look at this. When I explained the Malthusian argument with graphs (see here and here), this was one of them:
The horizontal axis is population size rising from left to right. The blue line is what I call excess fertility, which I define as fertility minus two. The red and yellow lines (where the yellow one is on top of the red line on the left) stand for what I call attrition. It is the product of fertility and mortality. You can think of attrition as the number of children born with the respective fertility who do not survive to fertile age. The difference between the two lines is the number of children born who survive to fertile age minus two, ie. how many more than two there are.
The definitions are perhaps a little obscure, but the graph is easy to read. The distance between the two lines is basically the growth rate up to a factor (some technical problems because fertility and mortality might not be at the same time, but this is only meant for illustrative purposes and as approximate). Where the two lines intersect, the growth rate is zero and the population stabilizes. Beyond that the growth rate becomes negative and the population shrinks.
Now, in this graph, the red line stands for Malthus’ explanation of a population “when unchecked” with a constraint for the food supply thrown in. Mortality, and with it attrition, suddenly jumps up at the maximum size. The yellow line is for a population where mortality already starts to rise before the maximum size possible without such extra mortality. There are two regimes here before and after mortality begins to rise. Before it happens, the distance between the two lines is fixed, hence we have a fixed growth rate and exponential growth. But when the yellow line rises, we have exactly the situation as above: the growth rate is tuned down linearly. So after mortality begins to tick up, population size follows the logistic function as above.
From this vantage point, it is clear that mortality could also rise in other ways than linearly. That would lead to similar “sigmoidal” functions in a more general sense. Another way to play around with this would be to add time variation for mortality, which might depend on a varying maximum size for the population without it, eg. because the food supply changes over time. I will not pursue this here because it does not lead to very different conclusions.
Verhulst basically brought Malthus’ intuition into a mathematical form that mortality could already increase before a maximum size for the population is reached . He assumed a very special functional relationship with population size via a linear transition. As noted, that is not the only possible way, any other function, eg. as the yellow line in the graph, where mortality increases from a minimum to a maximum would also be plausible. Despite the mathematical precision, logistic growth is just the Malthusian model with the “positive checks” in the special case that those lead to linearly rising mortality with population size.
There is nothing wrong with this. However, as I have explained in a previous post, it does not lead to a much better model. A population now no longer bumps into the constraint for the food supply in an instant, but only does the same thing more smoothly. It still has to grow to the maximum size that is possible. Since there are other sources of mortality, the maximum size is below what it would be without extra mortality. Yet, at the maximum size (or close to it because it is only reached asymptotically, ie. as time goes to infinity), the situation is as dismal as before. People still have the maximum number of children of whom all but two die until fertile age, just not only from baseline mortality, starvation and famine any more, but also from other causes.
This is the usual interpretation of the logistic model because people talk about the maximum size as “carrying capacity,” which means the population simply cannot grow any further. However, you also get logistic growth when fertility is variable as the population grows, ie. when there is Malthus’ “preventive check.” In my previous posts, I had this graph where both fertility and mortality change as population size increases:
The purple line again stands for excess fertility (fertility minus two), which is now variable and goes down from some point on with more population. The yellow line has the same mortality as above in the background. However, since the graph shows attrition, the product of fertility and mortality, it goes down also with fertility, not only mortality. Fertility starts to decrease earlier in this example than mortality, which is not necessary, only how I chose it.
On the left, we have a fixed distance between the two lines, ie. a fixed growth rate and exponential growth. When only fertility changes and goes down, we have the same case as above only the other way around. The growth rate decreases linearly, however now because of fertility and not because of mortality. That’s where we then have logistic growth, though only for some time.
When mortality also starts to increase linearly, attrition grows quadratically because it is the product of two linear functions. You can see this in the graph. The yellow line bulges a little on the right. The growth rate, visible as the distance between the two lines, decreases further, but not linearly. Hence we would get something like logistic growth, but not exactly, until the population stabilizes at the size where the two lines intersect.
Logistic growth is possible also here as we approach the maximum. But then mortality and fertility would have to rise towards higher populations sizes so that the difference between the two lines decreases linearly. I am too lazy to work through what that means because it is only an exercise to keep the logistic model intact for no reason. If you don’t do this, you lose the elegance of the logistic model above. But if you are willing to relax logistic growth to something “sigmoidal,” you already have this in the example.
The crucial point here is that the size at which the population stabilizes is no longer what should be called “carrying capacity.” The population can grow larger if it increases its fertility. This is not some maximum that cannot be breached. However, the usual interpretation for logistic growth is that it is about growth to a maximum size, “carrying capacity.” That would only be the case in the example before when fertility is at the maximum and the population just doesn’t have leeway on the upside.
— — —
All in all, I would view logistic growth just as a variation on the Malthusian model. The main difference is a smoother approach to a maximum size or the size that a population chooses to stabilize at with its fertility behavior. That is certainly more plausible than a hard constraint where the population bumps into a wall of steeply rising mortality, which is certainly a plus for the model. But it does not lead to any further insights.
A major problem, I see here is that the logistic model takes population growth as the relevant quantity whereas it would be better to derive it from fertility and mortality. Population growth lumps them together and makes a separate analysis impossible. If you are a Malthusian where fertility can hardly deviate from the maximum possible that is perhaps no great loss. But if also variable fertility might play a role and the population can have control over it, then this approach is awkward to say the least. But that is not a specific critique against the logistic model, also many others treat it as obvious that population growth is the relevant parameter, eg. the exponential or the hyperbolic model.
Hyperbolic Growth
The idea that a hyperbolic functional form might be more appropriate for population dynamics is more recent. It probably started with people looking at long-term data for humankind. Estimates for the distant past are extremely unreliable. Still as it seems, population growth on a very coarse scale was not with a constant rate, but with a rate that went up over time. This is actually growth that is even faster than exponential growth where you have a constant rate. Sorry, Malthusians, the exponential function is not the pinnacle of growth.
A function that shows a behavior with a growth rate that goes up with size is the hyperbolic function, which is just 1/x. Here is a graph where the axes are at 0 (picture from Wikipedia):
The hyperbolic function grows very fast at 0, to minus infinity when you approach from the negative side, and to infinity if you approach from the positive side. This behavior is called a “singularity” or a “pole.” Compare this to the exponential function: It needs infinitely long to go from 0 to infinity, all the way from minus infinity to plus infinity. Here it only takes “half as long.” It is hence intuitive that growth has to be much faster close to zero than for the exponential function.
Now, to apply this to population growth, you have to look only at the left branch, ie. when you approach 0 from the negative side, and you have to multiply by -1, ie. you take -1/x, which flips the graph around the horizontal axis and growth goes now to plus infinity. The rate of growth, the percentage change (technically: the derivative divided by the value) is the function itself. To see this take the derivative of -1/x, which is 1/x², and then divide the latter by the former, which yields -1/x again.
The rate of growth for the exponential function is a constant, its derivative is proportional to its value. The rate of growth for the hyperbolic function times -1 is proportional to its value. That means even the ratio of the derivative to the value goes up, which explains on an intuitive level why the hyperbolic function is so strong (it is not immediately clear, though, why it would have a pole).
Hyperbolic growth will be a better explanation than exponential growth whenever the rate of growth indeed speeds up over time. And at least on a very coarse scale that is what happened for humankind over time. You had a very long time of almost stagnation, perhaps from 100,000 BC until 10,000 BC, ie. a rate of growth hardly distinguishable from zero. Growth became faster with the neolythic revolution, before all the domestication of animals and plants, and eventually there was another speedup in modern times.
As far as I can see, the first to note this were Heinz von Förster, A. M. Mora, and and L. W. Amiot who called the hyperbolic function the “Doomsday equation.” I love the detached scientific attitude of this. In an article, published in “Science” in November 1960 (132 (3436): pp.1291–1295), they drew the conclusion that if the hyperbolic growth they thought they had detected in the data were to go on, the size of humankind would hit the singularity, and pretty soon, or as the ironically precise title states it: “Doomsday: Friday, 13 November, A.D. 2026. At this date human population will approach infinity if it grows as it has grown in the last two millenia.” I am sure you get attention with such clickbait.
Many others have pursued a similar approach, eg. the Russian multi-scientist Andrey Korotayev. According to his analysis, not only did humankind grow hyperbolically until about 1970, but GDP had even faster growth, which is supposed to be quadratic-hyperbolic or like 1/x² on the negative branch. That is a function again with a singularity at 0, but it goes to infinity from both sides. The implication is that GDP per capita (the ratio of GDP and population size) must have grown hyperbolically, ie. with an increasing rate proportional to population size.
In other words: Humankind grew and became richer at the same time, which completely contradicts Malthus’ pessimistic outlook where nothing can ever get better. Hyperbolic growth yields better fits for both the data for population size and for GDP, and so this is certainly an interesting candidate for population dynamics. But then I have some reservations.
The first is that as far as I understand it, proponents of hyperbolic growth like Korotayev are basically Malthusians nonetheless. It is curious that you run into the proof that the Malthusian prediction is completely wrong, but do not wonder whether the theory might not be wrong altogether. Instead Korotayev simply assumes that the Malthusian argument is correct on the micro-level. He only thinks that more population leads to more innovation, and even so much of it that GDP per capita can grow along with increasing population. That is basically a variant of Cornucopianism where humankind just won a race against inevitable and fast population growth (see my post on why I think that is silly).
But why would you expect that? The argument turns around the claim that a larger population will be more innovative. Maybe so. However, that is not enough: There is also an equivalent of death for innovations. New ones replace older ones or even make them counterproductive. And it is not clear that all innovations actually lead to more GDP. Think of a more efficient way to exterminate people or wage wars. Only if the net effect of innovations raises productivity will GDP per capita increase. But why should that happen in theory and even in such a regular fashion? Beats me. (I will address this point more in detail in a further post.)
Still, even despite this implausibility, the empirical observation is sound: GDP per capita broadly went up with population size. So arguing backwards, this is a possible explanation. To give you a hint how it could be otherwise, though: If productivity goes up first and then population grows slowly enough that also GDP per capita keeps rising, you would have the same outcome. But then there would be no race here. If there were no improvement, there would be no population growth. All it takes is progress at a certain pace and population would follow along, but slowly enough. Yet, that would mean that you reject the Malthusian model also on the micro-level because this implies control over fertility and with it population size. However, if you are a Malthusian at heart, I guess that is to much to ask of you.
There are other problems with an explanation via hyperbolic growth. For example, it leads to absurd conclusions both for the past and the future (granted the exponential model is just as bad). If you go back in time, population size has to go down to zero. And when you go to the future, population size has to explode at some point and go to infinity. Both cannot be true. Hence people like Korotayev need a fix. He introduces a deus ex machina, a solution that comes out of nowhere, that makes its appearance around 1970 and suddenly stops population growth against the natural law that held for all of history before that.
The usual culprit for bizarre behavior that goes against natural laws are women. More education is supposed to have led to a drop in fertility, and that was it. I will develop the argument elsewhere, but in my view, this is perhaps a coincidence. You have slowing population growth for other reasons and also rising female education, and then you draw the conclusion that because there is a correlation, it is probably causation.
If at all, the causation works differently in my view: Slowing population growth with continuing economic growth means more GDP per capita. Beyond some point this goes almost entirely into things that are not necessities, but in a way luxuries. One of them is education, also for women. The onset for both phenomena is so close together and the developments overlap so much that it may seem as if the causation runs from female education to falling fertility (not to exclude that more it can modify the development, which is not the same thing as causation).
Another problem in my view is the estimation. If you run a hyperbolic against an exponential model, the former must win by default for a very simple reason. As noted above, population in the remote past, let’s say from 100,000 BC until 10,000 BC must have been exceedingly slow, almost with a rate of zero. If humankind had continued like this, it would still be quite small, perhaps a few million. Since this is not so, and there are only 10,000 years to reach current numbers, growth must of necessity have gone up.
If you had only these two points: a growth rate of almost zero for ninety millenia and a higher growth rate for the last ten, then an exponential model with a constant rate will always do worse than really any model with a rising growth rate at larger population sizes. You can both accomodate the long flat part, and the increase at the end. Hence hyperbolic growth must win against exponential growth no matter how exactly things developed on a finer time-scale (assuming no big swings between data points and the like). The data matter only in as much as they have this general form, ie. only very little information goes into the analysis. It is fine to reject the exponential model, but it is not clear that that proves it is the hyperbolic model.
You also have the same pattern for the past two thousand years that von Förster, Mora, and Amiot looked at. In the first millennium, world population hardly grew, maybe stagnated according to some estimates or even shrank for some time. Most of the increase came in the second millennium, and there only very late. You could view this and the previous case also as stagnation followed by a phase of strong growth where you indeed have a speedup. However, that could be just two different regimes and not one development with the same dynamics. And the general pattern could be one of long stretches of stagnation and some growth spurts in between until a new plateau is reached.
The estimates for humankind over the first ninety millennia cannot help you with pinpointing the underlying dynamics also for another reason because they are extremely unreliable. That is critical when you try to find almost imperceptible growth that is only marginally different from zero.
If you try to fit an exponential model to the long-run development for humankind, it will lead to a very low rate of growth. That is the case because exponential growth is so fast and humankind could have only growth by a factor of roughly 30,000 over 100,000 years (from perhaps 300,000 to 9 billion people). Otherwise the population size would be gigantic. The average growth rate here is extremely close to zero, like 0.01% population growth a year. In a population of 300,000 people, as possibly 100,000 years ago, this is just 30 people more per year. In effect, such a population has to practically stabilize and add people on a case by case basis. And you would have to be able to get data that can tell you whether that was really the case.
Now, hyperbolic growth has a rising rate going forward, but also a falling rate going backward. This means that growth for the distant past in the model must have been even lower, practically 0%. But then how do you distinguish this from just stagnation? How could humankind ever have such marginal growth over the long run, but not simply have stabilized at some level? If so, how do you know whether hyperbolic growth was actually the case for most of human history? You would have to have extremely precise data to make a judgment here. But if it is unclear whether the model tells you anything about 90% of human history, what does it tell you then at all?
The flipside of this is that the whole estimation comes almost entirely from the recent past, and the impact of the data grows as you approach our times. A few data points over the past one or two millennia will have much more of an effect than the eight millennia before, and a few data points over the past centuries for the last two millennia. Effectively, the estimation only tells you something about what happened with the move to modernity or at most the past few millennia.
Maybe that worked like hyperbolic growth, but it is courageous to postulate this as the general behavior for all times, both going backward and forward. If the dynamics could have changed, and they must have done so unless you think humankind started out at zero, the generality is out the window. The extra theory that gets tacked on for the development over the past few decades is a telltale sign for a “general” theory that needs fixes at critical points. Again, an exponential model does not get around this either. But then both may be deficient.
As already alluded to above, an even deeper problem with the hyperbolic model are the data in my view. This is only mitigated somewhat by the fact that data in the more remote past do not play a major role anyway. Realistically, we don’t even have reliable estimates for world population before 1900. It becomes more and more fuzzy as you go back in time. There is perhaps a modest chance of getting a grip on data a few centuries back, maybe even for the past two or three millennia. But then it becomes very murky. Estimates for GDP are even shakier. If you then take the ratio of these two unreliable quantities to calculate GDP per capita, you may have more noise than signal when you analyze the distant past.
What makes it worse is that everybody expects to find exponential growth. Let me demonstrate what that means. Here are two estimates for world population (I take the underlying data from this Wikipedia page):
The blue line is for data from HYDE (2010, no citation in Wikipedia) and the red line is for data from McEvedy & Jones (1978). I only plot them until 1600, and the vertical axis is for millions. Note also that there are gaps that are automatically filled with linear interpolation in the graph, which means nothing.
As you can see, the estimates are very different, especially the farther you go back in time. And it is worse than it seems because the scale only shows absolute differences, which are smaller for smaller population sizes although the deviation as a percentage can be much larger.
That’s why I now plot the percentage deviation for the data points that are in both series (not all of them!). Note again that there is linear interpolation in the graph that may not be meaningful:
Most of the time, the blue line lies above the red line by 100%, at the peak even by 300%. Only after 1 AD is there moderate agreement, but still with a percentage difference of 10% or so.
And that’s only for these two data series that have estimates at many points back to 10,000 BC. If you look at all the data sets I referred to, you find even greater disagreements. As for 10,000 BC, there are estimates of 2 million, 4 million or 1 to 10 million people. In 5,000 BC, it could have been 5 million, 18 million or 5 to 20 million people. Even for 1 AD, the range goes from 170 million to 300 million (I exclude the Tanton data here).
Now connect this with the question of estimation. With hyperbolic growth you have very slow growth in the past, almost indistinguishable from no growth at all. What you think you are able to do is detect this in data that can move around by 100% or more. And that is only for population size, for GDP it is probably much worse. I have not looked into this.
Plotting the data on a logarithmic scale reveals an even more serious problem that I alluded to already above. A logarithmic scale is good to see the rate of growth, it is just the slope of the transformed curve:
The values for the blue line fall almost exactly on a line until 1 AD (I guess that the quibbles at the start have more to do with rounding for just a few million). What this means is that the data series, which apparently consists of many estimates, actually comes from just two data points that are connected by a straight line. But a straight lines means a fixed rate of growth. In other words, the authors “knew” that only exponential growth is possible and interpolated with that. As I argue in other posts, exponential growth is a poor model for what populations really do. So there is practically no information in the data points between 10,000 BC and 1 AD. It is only an assumption informed by a Malthusian bias towards exponential growth.
As for the red line, you see the same problem. The authors also “knew” that it could only be exponential growth. However, they seem to have four data points instead of two, another somewhere between 5,000 BC and 4,000 BC, and then one at 1,000 BC. They interpolate between the data points exponentially. There is not much more information in this data series than in the other one although this is obscured on an absolute scale. The authors seem also dissatisfied with the sudden breaks for the growth rate, which are, of course, not founded on any evidence. So what they do in addition, is add a little arbitrary smoothing to round things out.
I have not looked into the corresponding data for GDP, but my hunch is that you find the same thing. People assume that it will grow exponentially, make two or four estimates over 10,000 years and then interpolate with exponential functions. Maybe it is even worse: The estimates are for GDP per capita first, and then they get multiplied with population size to arrive at GDP, ie. you multiply errors here. If so, it is clear why the two data series move in lockstep. It is built into them because of assumptions, not actual evidence.
Now, imagine someone uncritically runs a statistical analysis on these data series supplied by someone else, especially if both are Malthusians and expect exponential growth. He will not be surprised to find it. But it is just what the other one plugged in who constructed the data series! This is a circular result. One “knows” that it must be exponential growth most of the time and fits the data to it, and the other “knows” that it must be so and finds it. The result is totally spurious then.
That causes serious problems for people who find exponential growth in this way. But it also has an impact on claims about hyperbolic growth. As you can see, the red line operates with different rates of growth. There is apparently some attempt to smooth the growth rate out. It perhaps felt weird to have a sudden change. Now, when you analyze such a data set, you will find that the growth rate went up smoothly although there is no underlying evidence for this. It is all just an artifact. When you then fit your hyperbolic model that looks for a smooth increase for growth rates, you will, of course, find it, and it will also have a good fit.
Not to be misunderstood, I do not doubt that there was an increase in the growth rate on a very coarse scale or even over the past centuries. It must have been so because you first had practically no growth for ages, and at the end you have much more. That makes hyperbolic growth more plausible than exponential growth from the start. But then these are not all possibilities, and you cannot argue by exclusion. Maybe the development was not as continuous: There were phases of stronger growth, but also plateaus between them. With only two or four data points behind the data, that is entirely possible. It might be so that later growth phases were indeed stronger. However, then we speak about them, not the general behavior as driven by population size at any point in time.
My conjecture is different how it worked. I will develop it in another post. Here is a short sketch: There were basically only two major events from 10,000 BC until perhaps 1,000 AD or even later: the domestication of animals and the domestication of plants. Both led to a ramp-up for population densities that were possible, roughly by a factor of 10 for each, and a factor of 100 for both together.
Those increases in productivity induced population growth commensurate with progress, but maybe slower than the increase in productivity, so GDP per capita could also go up. Each step led to massive population growth when it happened, but then there was a plateau or maybe only a slow further increase with smaller improvements. The fast growth happened on the local level and then spread geographically. That should have led to quadratic growth most of the time (see my posts here, here, and here), which also has a rate of growth that goes up with population size. The reason overall growth was very slow was that it was a combination of almost stagnation (at least by comparison) and very fast growth that only occurred at a frontier and involved a small fraction of the total population.
If that is true, these two big leaps would lead to phases of strong growth and slow growth in between or even stagnation. The pattern would be an overlay of two big shifts upwards and a third one over the past few centuries with the transition to modernity. In reality, all this would be smeared out somewhat because it was not all synchronous and took some time to evolve (people domesticated animals and plants at different times and in different locations, the geographical expansion was slow, etc.). So all in all, it might have been a development with three smoothed out ramp-ups that also overlapped to some extent.
Here is a very schematic rendering on a logarithmic scale:
The blue curve is the logarithm of population size with hard jumps. Those have the same size, ie. by the increase for population is by the same factor. The actual development would be a smoothed-out version of this. Hence you should perhaps think of the jumps not as sudden changes, but as the “center of gravity” in time when the big leaps had their peak impact on population size. I have placed the first big leap for the domestication of animals at 8,000 BC, the second big leap for the domestication of plants at 2,000 BC, and the big leap to modernity comes comes from 1800 AD on (note that the graph goes to the year 2000 although the numbers only go to 1600).
My timing for the big leaps is debatable, but only for illustrative purposes anyway. I will address the effect below for a different choice, which is exactly my point. If I had to argue for the timing: The domestication of animals roughly occurred around the time. My intuition is that the innovation could spread very fast. The domestication of plants came only a few millennia later. However, its spread around the world should have taken longer, and that’s why I place the “center of gravity” later when full agriculture really began to have a huge impact.
Now, I have added a quadratic function as an arbitrary smooth model (this is not quadratic growth for population size, but the logarithm) you could fit to the step function. It has accelerating growth as the population grows. The function has to be convex (bulges downwards) because of the spacing for the big leaps, where the distance between the first and the second is greater than betwen the second and the third.
What you find here now depends on the sequence for the big leaps and the time-frame. If you don’t agree with my timing, you might want to move the second leap further to the left and closer to the first. If you move it far enough the form for the smoothed out step functions becomes concave (it bulges upwards) from the first leap on. Hence if you fit a curve to it only from perhaps 10,000 BC on, you would find decelerating growth. Or you would have the same result if modernity had taken 10,000 years longer to arrive. If you took a longer time-frame where you have a lot of data points that are flat on the left, you can force a fit to be convex again.
My point here is: If the underlying development is driven by these three big leaps, the conclusion depends on their timing and the time-frame of your analysis. You can get accelerating growth and you can get decelerating growth. It could just be so that history worked out that it looks like self-accelerating growth with more population, and that looks like a hyperbolic model although there is no necessity about this.
Basically, that is also why Korotayev has such a problem with slowing growth lately: If humankind levels off afterwards, this will swamp the bias towards accelerating growth from the past over time. At some distant point in the future, a fit would have to be with a concave function, ie. you would have to conclude to the opposite dynamics: decelerating growth with population size. To avoid this, you have to argue this away as a weird exception.
— — —
Despite all these reservations, I see value in the hyperbolic growth model with its stress on some awkward facts for the Malthusian argument. Population growth could speed up over time (although I don’t see why it had to and why it could not also be quadratic growth), and even more devastating: population growth could go along with increasing GDP per capita, which should not be possible if Malthus were right.
However, the main problem seems to be the same as with Malthus and also Verhulst: It is the idée fixe that an explanation can only be a simple function that describes population dynamics once and for all. The mistake is to pose the question as: Is it an exponential, a logistic or an hyperbolic function that describes all of history? The simple answer could be: It is perhaps not an elementary function that you can write down explicitly as a power of x and with a few other operations.
I think this is a deformation professionelle of Enlightenment thinkers and also of those who are inspired by them. Thomas Malthus was a sharp critic of the Enlightenment, but he also naively took over its biases. For him, it is obvious that demographics has to be like Newton’s physics. Gravity decreases with the square of distance. And for demographics, there must be an equally simple function that captures population dynamics. That’s why Malthus is prone to latch on to the exponential function, which was apparently the only one he knew apart from the linear function.
— — —
So much for logistic and hyperbolic growth. I will address other approaches that have been pursued by Lemin Wu as well as Voigtländer and Voth. I see value there, too, because they grasp an essential point about demographics that Malthus missed: Not all economic growth has an impact on population dynamics. However, they then still take the Malthusian assumptions for granted and do not think them over. In the final analysis, this leads to models that solve some problems, but then create others with their explanations or the internal logic is strained.
