Is Quadratic Growth a General Phenomenon?

[This is related to 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?”]

If you are tired of my posts on demographics, relax: What I will develop below is not about population growth in the first place, but about economic growth and how to interpret it. To get going, here is a question for you:

Look at this graph. What do you see?

Many people will answer: This is exponential growth!

However, I’ve tricked you. What you see is a quadratic function. On an intuitive level you identify exponential growth in this way: The function goes up, and its slope also goes up on top, ie. the function grows faster and faster. Hence, this must be the exponential function! Yet, all the criteria also apply for the quadratic function: x². Its derivative is 2 * x, which means that the slope is always positive and increases as the function grows.

A further reason why I could dupe you is that I wrote about economic growth at the start and also population growth, so you probably expected to see an exponential function. Because it seems obvious that this is the go-to model.

But as I have argued for population dynamics: Quadratic population growth is a much more plausible model in a growth phase than exponential growth, see here, here, here, and here for more specific posts. Also the empirical data are regularly better in accord with quadratic than exponential growth. What I did not realize at first, though, was that this observation may have much broader implications that go beyond demographics, eg. when you are interpreting economic growth.

The difference between the quadratic and the exponential function is this: If you calculate growth rates, ie. you take the slope and divide it by the value, then for an exponential function, this is a constant:

You have for the slope: exp’( r * x) = r * exp( r * x ).

And after division by exp( r * x ), you always get r.

However, for a quadratic function x², the slope is 2 * x, and after division by x², you obtain 2 / x for the growth rate, which is a hyperbola that falls off to zero as you go to infinity. In other words, the exponential function always grows at the same rate, while the quadratic function has a growth rate that goes down to zero. Yet, farther away from zero, the growth rate decreases very slowly. It takes infinitely long to go to zero. Hence over finite intervals away from zero, this can be almost a constant, which may look deceptively like you have an exponential function. It is hard to tell. And hence it is not easy to identify which is which, especially not if you try to do it visually. And if you expect an exponential function.

— — —

A quadratic function seems to come out of nowhere. Why would you expect it in any context at all? This seems totally arbitrary.

However, there is a simple argument why you should expect it in many situations. I developed this first for the demographic case, but as I now realize that is not the only example where something like this might happen.

Suppose you have beings that live essentially on a two-dimensional plane, kind of like humans. This is an idealization because we live on a sphere, but as a first approximation it is not too far off. I will handle the problems in a moment that arise from this simplification.

Now, suppose also that there is some quantity of interest, in the demographic case: population density, but it could also be something else, and that quantity is at first at some lower level, but then goes up to a higher level. There are many ways how this could work out. An extreme case would be that the level just jumps up. Obviously, this is not possible for population density. Another way to connect the levels would be with a logistic function that speeds up to half the difference between the levels, and then slows down symmetrically as you approach the higher level from below.

As for population, I think it is something rather similar, but with a twist: There is population momentum. If you have additional children beyond the replacement level, then those will have further children and grandchildren down the line. Since generations overlap (most people live long enough to see their children and even grandchildren grow up), these additional descendants come on top. That means the population keeps growing for about half a century after fertility exceeded the replacement level, which can happen even if fertility goes down or below to the replacement level in parallel.

This lagged effect makes going to the higher level hard. Any decisions now have repercussions for decades. And so a population that tries to go to the higher level might overshoot it. It then has to correct its size downwards again, which has a momentum effect in the other direction, and then you have another round of overshooting and undershooting. If the population does it right, these oscillations will die down over time and the population size converges to the higher level. But basically, this is some kind of logistic growth only with delays from the past that make things sticky and introduce some oscillations around the higher level.

— — —

However the transition may work, this is what happens at one location. Now suppose that this shift from a lower to a higher level first occurs at one point, a center, and then expands with a constant speed over the infinite plane. It is particularly easy to understand what happens if you assume a sudden jump from the lower level to the higher level, not a more complicated function like the logistic function and even with delays.

At first the level jumps up at the central point. But then as this spreads out at a constant spead, you have two different regions. Either you have the lower or the higher level. The higher level applies for all points in a ball around the center with a certain radius. Outside you still have the lower level. Since I assumed a constant speed of expansion, the radius of the ball grows linearly in time. And this means that the area of the balls grows quadratically. You have two dimensions how it expands. Hence if you look at the aggregate difference, it is just area times the difference between the higher and the lower level. The latter is a constant (by my assumption), and the former is a quadratic function, so this is a quadratic function. And that’s what you should expect in such a situation. Not exponential growth!

It becomes slightly more complicated if you have a transition from the lower to the higher level that is not as simple as a jump. But then you can look at a line that runs from the boundary to the center. At the boundary, there was no time to go through the transition, so you are still at the lower level. However, as you move inwards, there was more and more time. Since the radius expands at a constant speed, you now see the behavior at a location work out as you move towards the center, only with a distortion because of a constant for speed.

If the transition converges to the higher level over time at a location, and the expansion has gone on long enough, you will have that, for points in the interior, the level is almost the higher level. The transition happens on a zone close to the boundary. It always has a fixed depth before the level is close to the higher one in the interior.

As the ball grows, more and more of it is close to the higher level. The transition zone has an area that grows linearly, It has a fixed depth and only grows like the circumference, a one-dimensional entity. Its area is hence only a linear function in time. But that means that the share of the transition zone in the ball falls to zero. Most of the ball is almost at the high level, and only a dwindling share is somewhere between the levels. All in all, this a somewhat smoothed out version of the case with a sudden jump. Mostly it is the same apart from a transition zone whose relative contribution goes to zero. Hence you also have almost quadratic growth in this case, at least after an initial phase with very small balls perhaps where the behavior at a location dominates.

— — —

From this vantage point, it is also clear why exponential growth is actually impossible for beings that live in two dimensions. It speeds up much faster than quadratic growth. However, you can only have that with a constant speed of expansion if the level not only goes to a higher value, but to infinity. Basically, you must have an exponential function divided by a quadratic function for the level (think of it as a jump) to get an exponential function in the aggregate. However, the exponential function is stronger asymptotically (as time goes to infinity) than a quadratic function, and so the ratio goes to infinity. Yet, if the beings need some minimum space, population density just cannot go to infinity.

The other way out you might try is to expand the balls at more than a constant speed. But then you must have that the radius expands as the square root of an exponential function. Only in this way you can get an exponential function if the level only goes up by a finite amount. However, the square root of an exponential function is exp( r * x / 2 ), which is an exponential function itself and hence goes to infinity. And that is not possible for beings either that have some speed limit. Also combining the two ways does not help you, at least one has to go to infinity.

To stress this point: Exponential growth is indeed very strong asymptotcally, but that means it is impossible for beings even under the most favorable circumstances who essentially live in two dimensions, need a minimum of space and can only expand at a finite speed. That is even so if you assume an infinite plane with an infinite area and no constraint for the food supply or anything else whatsoever. The whole worry of Malthusians about the asymptotic case is completely pointless because exponential growth is impossible for beings like humans already in theory.

If I tricked you with the quadratic function above, then don’t feel sad, in the context of populations everybody is tricked by this because Thomas Malthus messed this up with his obsession about the exponential function. You will get the regular claim that bacteria show exponential growth. But if they grow in two dimensions with a finite speed and some minimum for their size, eg. on a Petri dish, they have quadratic growth. Period.

There is a whole genre of bacteria videos, see here on Wikipedia for Escherichia coli alone. And you will have many people who look at these videos and exclaim: “Wow, this is exponential growth!” But pay close attention, the cell colonies expand at a regular speed, ie. what you actually see is quadratic growth. If bacteria can grow in three dimensions, the same argument leads to cubic growth, which is still much slower than exponential growth asymptotically. Only in a suspension where there is initially a lot of space and if you swirl the cells around so they do not get in the way of other cells, may you have exponential growth for some time. Until it is over.

— — —

To sum this part up: If you have a quantity that goes up from a lower to a higher level at one location, maybe with one jump or also in a smoother version of this, and the area expands from there, then you will find something that is close to quadratic growth. That must be the natural assumption here, not exponential growth.

You may object that there could be more centers. For example, some from the first colony go to a far away place and start their own colony. Note that this already means you fiddle with the speed of expansion. That can, of course happen. But as long as the two colonies remain separate, you then only have a sum of two quadratic functions, which is a quadratic function again. As soon as the two colonies grow together, it gets slower, not faster because they get in each other’s way.

Now, to address the point above that we are not on an infinite plane, but on a finite sphere: If we first disregard that there is even more to it, the whole geography, and view the planet as a billiard ball, expansion with a constant speed at first works almost like on an infinite plane. Locally a sphere is well-approximated by a plane. However, since a sphere has a positive curvature, the area in balls around a center expands more slowly than on an infinite plane. When the expansion has already gone from the pole to the equator, it becomes even worse. And eventually, when you reach the opposite pole, it is over.

However, that means that on a sphere, growth is even slower than quadratic growth, and hence it is still farther away from exponential growth. Since there is at best a finite area here, growth even stalls from some point on while on an infinite plane it can go on forever.

This is still not realistic because our planet is not a billiard ball, but has more structure. So expansion might already stop much earlier, eg. when you reach the oceans. However, that makes growth still slower. If you have already expanded in the upper half of a region like a peninsula and the expansion effectively is only in the orthogonal direction, growth becomes linear. And then it will slow down further until it comes to a standstill. Hence the actual behavior on a sphere plus a complicated geography is at best quadratic growth initially, will then fall to linear growth and fizzle out eventually.

The crucial point here is that what happens at a location does not matter a lot for the behavior in the aggregate, how you go from the lower to the higher level, as long as the speed of expansion is low. Only if the expansion went immediately to the whole world would you see the behavior at one location everywhere and hence also in the aggregate. If the speed of expansion is slow enough that most of a ball is almost at the higher level, then the effect from expansion dominates and is what you see in the aggregate. In other words you will have something like quadratic growth initially in a growth phase that later falls to linear growth and then runs out.

— — —

Now, I have discussed this for population density, which goes from a lower to a higher level. But then you could make the same argument for any quantity that also behaves like this at one location. This has nothing to do with what the quantity is as long as it has such dynamics. Take for example some innovation that makes more output possible, but a constant increase once, not some sustained growth. The transition might take some time. However, if the speed of expansion is slow, it does not matter all that much how the transition works out, and you will have almost quadratic growth in the growth phase and so forth.

Here is a funny application for this. Suppose we have only one innovation that yields a lot more output. Historically, that might have been something like the domestication of animals and then the domestication of plants. Part of the higher level might be that the innovation also induces a shift for population density upwards at a location, which boosts the effect from what it is per capita. And let’s assume all complications for the moment away, this innovation is perfect right at the start and needs no further improvements. At a location, it takes some time to go from the lower to the higher level, but perhaps not very long, let’s say half a century to accomodate also population growth to a higher population density. And then this innovation spreads across the world, but it creeps from one location to the next.

Although what happens is just one innovation and there is nothing behind it that drives output growth forward over time, you will then get almost quadratic growth for a long time. That is also very regular. It may look like exponential growth, so you may be tempted to look at the growth rates and interpret them as driven by many small innovations that come in over time and raise the aggregate at almost a constant rate. But that is not it at all here by assumption. What you see is the effect of geographic expansion. There are no new innovations that come in, it is only one big innovation at the start that expands over the world.

Now, with quadratic growth and even more so if geography drives it down further, you will find falling growth rates over time as you should. However, with an exponential mindset and an interpretation as many small innovations that come in, you will read this as: Innovation was pretty fast initially, but then it slowed down. The population here got less and less innovative. In a way that is true because there was only one big innovation at the start and then literally nothing. Yet, your understanding is misled by your assumptions. You will now try to figure out what slowed innovation down.

You can go in circles. Maybe there are other quantities that also expand in parallel over the world and that also have the same behavior. So you will find lots and lots of spurious correlations that look like an explanation. For example, you might study how many new businesses are started per capita. But if starting a new business, eg. setting up a new farm, for this one innovation also expands geographically, you must find that it goes down along with economic growth. For most of the area the businesses have already been started, and new businesses can only occur close to the boundary, which is a smaller and smaller fraction of the total area. You might be talking effectively about the same thing, and then it is no surprise you find a connection.

If you are at a loss, you can also tell a story about diminishing returns for further innovations. But by assumption, there are none anyway. You may also start to worry about the innovativeness of the population, tell “just so” stories how it losts its dynamism and so forth. All wrong. That is not what happens here. It is one innovation that expands geographically, and that leads to a certain growth behavior in the aggregate where the growth rate must go down.

Here is another application: Suppose that again you only have one innovation at the start, but the speed of expansion goes up, however, otherwise everything remains the same. Now, this will lead to faster growth. Yet, if you are into an explanation with many slow innovations that come in, you may get a completely mistaken idea: The population has found its old dynamism again and has become more innovative. Apart from perhaps one or a few innovations that speed up expansion, this is not the case. And the growth comes after that, maybe long after it. You will also observe that associated quantities like how many businesses are set up per capita go up along with economic growth, and pinpoint this as the driver. But you are again looking in the wrong place.

Surely, the world is not that simple with only one big leap. For example, you could have also two big leaps. But then this works out in the same way if expansion is the dominant contribution. You add two quadratic functions and it is still a quadratic function. For the same reason, an objection does not work that a big innovation might come in smaller increases. Of course, that can happen. But if the main part is geographic expansion that is still what you might see in the aggregate.

Or think of a situation where expansion was thwarted for some time. There were perhaps some big innovations, but they could not spread. Then you lift this constraint for them all at once. So it is like one big shift from a lower to a higher level. That should result in rather fast growth after a period of stagnation that then later slows down. Think of the Great Depression and World War II, and the strong growth from 1945 into the 1960s. Maybe the point here is not that people got more innovative at the time, only that there was a backlog of innovations that then could become effective as one big leap. In a way, this would be the economic equivalent of the “baby boom.”

— — —

The distinction between these two interpretations, exponential and quadratic growth, is particularly important if you want to extrapolate to the future, especially the distant future. The exponential function speeds up more and more. But a process that is basically by geographical expansion grows at best quadratically and must later even slow down and come to a halt. Over a short time period, the two models may look deceptively similar. But as I have demonstrated in one of my previous posts for the population of England, which one you pick can make a huge difference for forecasts.

What I did in my post is use a data series from 1815 to 1869 for the population of England. Then I did a fit with an exponential and a quadratic model. Note that already on the data the quadratic model works slightly better, but both look pretty good as an explanation. If you then use this to extrapolate and forecast population size in 2015, you get 145 million for the exponential model, which is ludicrously false. The quadratic model, which must be an overestimate because of the underlying geography, yields 69 million, which is at least in the right ballpark: England had a population of about 55 million in 2015.

The more fundamental point here is that it may be misleading to look at things via growth rates which make you biased towards exponential thinking. If the underlying process is more like a shift from a lower to a higher level and that expands in two dimensions, the natural model, at least in the growth phase, would be quadratic growth. Growth rates that slow down should not be a surprise then. And it is also futile to try and relate them to other quantities and hunt for processes that drive innovation down. That could be a pure artifact and in the worst case a correlation of geographic expansion with itself.

While it may seem obvious with an exponential mindset that there is always some force that drives things forward with a constant growth rate, actually it seems far more realistic to me to view economic growth as going from a lower to a higher level with some innovation and then remaining there unless you have another idea. If so, you cannot extrapolate as with an exponential function. It depends on whether further innovations really materialize, which may just depend on whether they are in the cards at the moment or not. People could domesticate animals, one species after the other. But from some point on you could not force new species for domestication into existence. And you could not do this with plants either.

The natural assumption from such a vantage point is that you see shifts from lower to higher levels that work out slowly over time, at first with quadratic, later with linear growth and then stabilize at the higher level. When there are further innovations that raise the level higher still, then you will go to the next plateau, and so forth. There is no reason to expect this to go on to arbitrarily high levels. Just as far as the innovations carry us.

Surely, you cannot forecast with that as effortlessly as with an exponential function because it is inherently unknowable what will happen regarding further innovations. But then it is perhaps realistic to grant this point and just abstain from unwarranted extrapolation. All you could do then with some justification is extrapolate those processes perhaps that are already under way, and only as a transition to a higher level, not as the start of exponential growth with a constant rate to infinity.