The Malthusian “Red Whale”

[This is the continuation of my series of posts on Thomas Malthus’ “An Essay on the Principle of Population,” first published in 1798, which began with “What’s Wrong with the Malthusian Argument?” and “Malthus and His Geometrical and Arithmetical Ratios.”]

If you have ever read Malthus or any of his disciples, one thing may strike you. They are extremely confident that their theory is so self-evident that only a fool could doubt it. I think a major part of this smugness comes from one point that Malthus makes. But as I will show now it is not just a “red herring,” but rather a “red whale,” a huge distraction from what is really at stake.

The trick is that you are led to believe that the Malthusian argument stands or falls with the truth or falsity of one claim. If you cannot refute it, then Malthus is right. And you cannot refute it. My hunch is that many people have tried this and then they concluded that indeed the theory is beyond a doubt. Not the least Malthus seems to have believed it, and that may explain his condescension with regard to any criticisms. At most he would concede that he had to fix a few details, but that would have no impact on his larger argument. However, this “red whale” is completely inessential. You can just accept it, and still refute the Malthusian argument.

What I mean is this (cf. I.17):

Assuming then, my postulata as granted, I say, that the power of population is indefinitely greater than the power in the earth to produce subsistence for man.

As Malthus tries to establish in side-arguments, he thinks that population “when unchecked” will grow exponentially, and that subsistence, which mostly means the food supply, linearly. Note, though, that the claim here is stated for population in general, and not “when unchecked,” something that Malthus himself concedes has never been the case for an actual population.

As I have already developed in my last post, this claim cannot follow from the two postulata that (1) the food supply imposes an upper bound on the size of a population and that (2) the “passion between the sexes” will remain “nearly in its present state,” whatever that means. The reason is quite simple: The assertion here is about two functions, an exponential and a linear function, and has nothing to do with what those functions stand for. It would be the same if you had “processing power for microchips grows exponentially” and “the running number for US presidents grows linearly.”

Malthus’ wording suggests that he thinks of some inherent quality of a function, its “power” and wants to compare the two “powers” for an exponential and a linear function. There is only one interpretation in which this makes sense, namely if you read it as an assertion about the relative asympotic growth behavior of the two functions, ie. as time goes to infinity.

You can also recast it in this way: If you divide an exponential function by a linear function, the ratio will go to infinity as time goes to infinity. But note, because this is important: The claim is about what eventually happens, not what happens right now or over a finite timespan. There is no contradiction that one function has more “power” than another, but is still weaker now or for some finite time, ie. the ratio goes down, not up.

Now, if you are not a mathematician, this may shock you: It is incontrovertibly true that an exponential function grows faster asymptotically than a linear function. This is a necessary truth that is not negotiable. And you can show this by supplying a mathematical proof. If you are not a mathematician — and Malthus himself has a very low level of sophistication even by the standards of his time — this may seem like a revelation. You rack your brain to refute it because this seems to be a central claim of the Malthusian argument. But no matter how hard you try it cannot be done. And this creates a feeling of extreme stringency. In this case it is true that only a fool could doubt it.

Since I assume you are not a mathematician, I will supply two arguments that are perhaps too sloppy for a mathematical proof, but give you an idea how one might work. The first explanation is rather intuitive, the second more technical. However, both are not very difficult to understand as you will see.

(1) Exponential growth means that the slope of the function is a fixed proportion of its value. The larger it gets, the more it speeds up. For fixed time periods, this is an increase by a percentage. Linear growth means that the slope of the function is a constant, or that absolute increments over fixed time periods are constant.

You can now understand the different “power” of the two functions: As they grow, the linear functions always has the same slope, whereas for the exponential function it gets steeper and steeper, and not only that, the slope goes to infinity as time goes to infinity. If you now take the difference of the two functions, its slope is the difference of the two slopes. One explodes, the other remains constant, so the difference may start at some value, but then it goes to infinity.

If the initial value for the difference between the slopes is negative, the linear function grows faster than the exponential function. This is just a complicated way of saying that it has a steeper slope. But then at some point, the difference hits zero because it goes to infinity, and the two grow at the same speed. And afterwards the difference is always positive, ie. the exponential function grows faster than the linear function. Note that before the difference of the slopes hits zero, the linear function is stronger. You can also see this if you go to minus infinity. The linear function goes to minus infinity while the exponential function only falls to zero. Malthusians are so obsessed with the strength of the exponential function that they tend to miss this point. In the other direction the exponential function falls off extremely slowly. But you can see many Malthusians who argue as if it falls off extremely fast.

After the difference of the slopes has hit zero and is always positive, the exponential function gains on the linear function and that even speeds up more and more. So after some time, it will catch up with the linear function although it might have started well below it. And from then on, it will shoot past it. The critical point here is that this is an asymptotic result, ie. as time goes to infinity. There must be some point in time from when on the exponential function leaves the linear function behind.

But before that the exponential function is below the linear function, and before the difference of the slopes has hit zero, the linear function grows faster than the exponential function. Hence you cannot make any general claims like say: The exponential function will always be greater than the linear function in 25 years or even immediately. It only holds as time goes to infinity. Again, if you watch their hands, Malthusians often try to deal you something else: We know it is an exponential function, so it will beat anything right away.

The point in time when the exponential function leaves the linear function behind depends on the parameters: for an exponential function exp( r * t ), this is the rate r, ie. the ratio of its slope and value, and for the linear function its slope a and intercept b, in a * t + b, where t is always the parameter for time.

For a given rate, and a finite timespan, you can always pick a slope and/or intercept for the linear function that makes it greater. Just raise the intercept and move it upwards and/or make it steeper by increasing the slope. Of course, if you want to make the comparison you cannot fiddle with the intercept and slope afterwards, but have to keep them fixed. However, since this is so, you cannot prove the claim over finite timespans that an exponential function will be greater than a linear function in any case. It is simply false. For some parameters the linear function is greater, for others the exponential function, or it might change in between.

(2) If you are not into mathematics, you can skip the technical argument that follows. It goes like this: You can write the exponential function as an infinite sum of powers, eg. as a Taylor series around zero. Since with each derivative another factor r, which is the rate, drops out, you have constants before the powers that are all positive if the rate is positive. If it is zero, the exponential function degenerates into a constant function, if it is negative, it falls off, something Malthus never considers with the generality of his “proof” that does not exclude these cases.

The linear function has also an expansion as an infinite series, but it stops after just two powers: t⁰=1 and t¹=t. Now, higher powers of t grow faster asymptotically than lower powers, just take their ratio, which is still a strictly positive power of t and which hence goes to infinity. And since the exponential function has powers for any exponent to infinity and those have all positive coefficients if the rate is positive, it has contributions that grow faster than a linear function. But then this also holds a forteriori for the infinite sum. Actually, you can show much more: The exponential function will also beat any polynomial, a finite sum of powers.

If you are a pedantic mathematician, my “proofs” are not yet rigorous enough. But that can be fixed. However, my guess is that if you are a mathematician you will accept the claim also as a truism and exclaim: “Duh!”

As I have noted above, you cannot prove the claim about the relative asymptotic growth behavior over finite timespans. Cringeworthy as it is, Malthus still thinks he can do it. Since he confuses the proof about the relative power of exponential and linear growth with the question what the rate and slope of the two functions are, he thinks of specific parameters in both cases (cf. my previous post). If you make this assumption, of course, you can demonstrate the claim over a suitably chosen finite timespan. But that is not a conclusive proof for the general result that Malthus wants to show. He still spends a lot of time on making his case in this way, which has the rhetorical advantage that his readers focus on it although it is inessential for his general argument. As I said, this is a “red whale,” and that goes a long way to understanding why Malthusians are so smug.

If you think I am asking too much of Malthus. First, this is not about mitigating circumstances for something that goes wrong. The question is whether his claim is true. And second, the exponential function had been studied by mathematicians for over a century. It appears quite naturally when you try to understand compound interest. The brothers Jacob and Johann Bernoulli had begun to analyze it in the late 17th century. The base ‘e’ is named in honor of Leonhard Euler (1707–1783) who had died before Malthus wrote his book. And Malthus works with some of Euler’s tables in the later editions. So he could have had a better understanding. However, he did not.

Instead, Malthus works through a numerical example (cf. II.14). He starts from seven million inhabitants in Britain at his time, and then doubles this number to obtain a progression of 14, 28, 56, and 112 million. The steps are for 25 years, so this is over a century. He then sets up a second progression which starts with food for seven million people, and grows by fixed increments for food that can provide for seven million people, which gives food for 14, 21, 28, 35 million. Obviously, there are way more people after the first step than food. This is scary, but also absurd because it is simply impossible that after half a century there are 28 million people with only food for 21 million, and then this goes on for another half century with an even greater discrepancy. Of course, you get a feel for how fast the exponential function increases, and that it cannot go on long before there is a mismatch. The claim is indeed true for the asymptotic case.

However, over finitely many steps, this is not the consequence of linear growth, but of Malthus’ choice for the fixed increments. If Malthus had taken not food for seven million people, but for 27 million people as the fixed increment, he would have obtained a progression for food that can provide for 7, 34, 61, 88, 115 million people, which is also a linear function, but where all people are provided for over the century. It might be objected that Malthus has supplied an argument that the food supply can only grow with increments for seven million people. But then the reference point is arbitrary. If he had assumed a reference population of 27 million people, the same argument would have led to increments for 27 million people. To wit, that is an argument that Malthus’ critics already pointed out to him at the time.

Next, Malthus senses a possible objection that people could emigrate, and that this would keep population size and the means of subsistence in line. He tries to get around this with an argument that emigration is painful (cf. II.15), which is sorely inadequate in a mathematical argument. It is not the question whether people are unhappy when they have to emigrate, but whether this can keep population size and the means of subsistence aligned. And, of course, it can. It only shifts the problem elsewhere, though. But as long as there is still uncultivated land, it can postpone the day of reckoning.

Malthus, therefore, now excludes an open system with emigration, and goes over to a closed system where this is not possible: the whole world (II.16). His claim about the size of the fixed increase, the slope of the linear function, comes from his derivation for Britain circa 1800, a mostly settled country. But he now also assumes the same increase, namely by a fixed amount equal to the initial total, for the whole world, which still has plenty of uncultivated land as Malthus concedes elsewhere.

As a supporting argument, he points out that the specific linear function he has in mind goes to infinity (cf. II.16). But any linear function does that, and this does not imply a specific slope, which is the gist of his claim. He then sets up the same example again, now only starting with 1 billion, not seven million (cf. II.17):

Taking the population of the world at any number, a thousand millions, for instance, the human species would increase in the ratio of — 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, &c. and subsistence as — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, &c.

Of course, this yields a disproportion of 4096 billion people to only food for 13 billion after just three centuries. But again, for finitely many steps, this does not hinge on the linear increase for the means of subsistence, but on the slope which Malthus assumes. With more steps, the accelerating growth of the exponential function becomes easier to grasp. So, as an illustration, this is fine. And asymptotically, the argument is correct. For more and more steps, you would have to assume ever larger slopes to keep population and food in line. Still, this is not a proof, which would require a mathematical argument that Malthus seems unable to make.

Now, why is the claim for the different “powers” of an exponential and a linear function inessential for the Malthusian argument even if it were established with a mathematical proof? It is indeed incontrovertible, but that is besides the point.

The simple answer is: Because it is an asymptotic result as time goes to infinity, and Malthus wants to draw conclusions for finite timespans and even for the situation at any point in time, ie. the infinitesimal case. That becomes clear when he finally states what he is after (cf. II.18):

[…] yet still the power of population being a power of a superior order, the increase of the human species can only be kept commensurate to the increase of the means of subsistence, by the constant operation of the strong law of necessity acting as a check upon the greater power.”

That’s a non-sequitur, a conclusion that does not follow from its premises. What Malthus has shown at best is that population grows exponentially “when unchecked” and subsistence linearly. I will show that he has not even done that. But if we grant it anyway, he can only conclude that asymptotically the size of an unconstrained population will grow faster than the constraint dictated by a linearly increasing food supply.

Malthusians rightly sense a tension here. But it is a tension that arises because Malthus assumes two contradictory claims: a population is not constrained and it is at the same time constrained. In addition, he has conveniently forgotten to mention the “when unchecked” that he had earlier in his derivation. But that means that he skips from what he thinks has never happened to the general case. However, that is just a trick.

Now, if the population is constrained, it cannot grow exponentially as time goes to infinity. That is impossible. So the asymptotic case is completely irrelevant. All the numerical examples, he has supplied are only a huge distraction, apart from the faulty logic as a proof for his claim: not just a “red herring,” but a “red whale.”

Malthus seems to think of the “powers” of an exponential and a linear function as an inherent quality that applies always. Yet, as I have explained above, that is not true for finite timespans or even infinitesimal timespans, ie. at any point in time.

His invocation of the “strong law of necessity” is another confusion Malthus introduces into the argument. It is necessary that an exponential function grows faster asymptotically than a linear function. You can supply a mathematical proof for that. And it is also necessarily so that you cannot have it both ways: a population is not constrained and it is constrained. Both are basically logical necessities. Still, what Malthus insinuates is that there is some natural law that always operates on human populations at any point in time, which is something entirely different.

Of course, there is something that Malthus has in mind here: The constraint is binding. If a population has already hit the upper bounds dictated by the food supply, it would grow beyond it if it were not constrained. The population might ignore this and behave as if it were not constrained and have just as many children. However, even then that is only true, if the size of the population is already so large that the slope of the exponential function is steeper than the slope of the constraint. Otherwise the constraint leaves it behind and it can grow exponentially as the constraint passes it by.

If the slope of unconstrained growth is steeper than the slope of the upper bound for the constrained population, then infinitesimally the population tries to grow beyond it. In that case, there is not enough food, which means some will die indirectly or directly from starvation, and that will dip the slope of the constrained population down to the constraint from the food supply.

This is indeed a harsh situation, but it a very different necessity than the one Malthus could establish in his derivation for the “powers” of an exponential and a linear function. It is the necessity of his first postulatum that “food is necessary to the existence of man” (cf. I.14). But that was his claim from the start, and so the whole discussion about the “powers” does not add anything beyond that.

And then the point is not about the asymptotic behavior of two functions, but about their slopes at some point, ie. the infinitesimal situation, which is as far removed from the former as is possible. Any other functions that have such slopes at some point will lead to the same result (up to higher orders), no matter what they do as time goes to infinity.

Take a linear function with a steeper slope for population growth than the slope for the linear upper bound dictated by the food supply. Both functions now have the same “power” because their ratio goes to a finite constant and not infinity as time goes to infinity. Still, by the same argument as above, the steeper linear function will catch up with the flatter linear function at some point in time. Once it has, the infinitesimal situation at that point is the same (up to higher orders) as for an exponential function. So the result does not hinge on its asymptotic properties at all, only on the fact that its slope exceeds the slope of that for the linear upper bound.

Malthusian thinking is a hardy weed. My guess is that its proponents would be little impressed with my arguments. Yes, true, Malthus’ derivation has more holes than a Swiss cheese, but still he gets it basically right, and his insight is correct. As I will explain in further posts, this is not so. Malthus’ sweeping claim is that human populations regularly find themselves at an upper bound dictated by the food supply, and that they ignore this and keep having children at too fast a pace. If that is so, then indeed always some will starve to death because it is not possible otherwise. However, the crucial point here is to establish that this is indeed regularly the case.

Malthusians gloss over this, and will point out that famines have happened in the past or that people have often died of starvation. Unfortunately, this is so. But that does not imply that they are regularly in such a situation or that this is because of a binding constraint for the total food supply. You can also have this if some people lack the means to buy enough food although it would be there. And if the food supply suddenly crashes because of some unexpected catastrophe — a drought or when a volcano erupts around the world — there is then not enough food and that has dire consequences. But that only shows that this can sometimes happen. And it is a simply fallacy to conclude from some examples to regularity, ie. that it is always so. But then that is the claim Malthus is after.

— — —

Here’s an overview with all the articles and related ones in this series. There is also a short summary for each post, so you can follow the argument even if you don’t feel like reading everything. I will keep the list updated:

Synopsis: What’s Wrong with the Malthusian Argument?