Subsistence Increases in an Arithmetical Ratio?

[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 posts here that I will keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?”]

One of the central claims that Thomas Malthus makes in his “Essay on the Principle of Population” is this (cf. I.18):

Subsistence increases only in an arithmetical ratio.

The “only” is meant as a comparison with different dynamics for population growth. “Subsistence” mostly stands for the food supply here, and an increase “in an arithmetical ratio” corresponds to linear growth on a continuous time scale. So in modern parlance the claim reads:

The food supply grows linearly over time.

Malthus makes it sound as if this conclusion follows from his two postulata. This could only be so if he could derive it from the first one (cf. I.14):

That food is necessary to the existence of man.

However, the most you could conclude from it is that there is an upper bound for the size of a population with a given food supply. But there is no way to show how the food supply develops over time, even less so that it has to grow linearly. This would be a complete non-sequitur, a conclusion that does not follow from the premises.

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Malthus attempts to supply an independent argument later on in his exposition. The question is whether the food supply grows linearly. But confusingly, he mixes this up with the unrelated and rather inessential question what the slope of this linear function would be. With this shift of focus, his readers are distracted from the main point that has to be established: the functional form.

The argument proceeds in this way: Malthus starts out by claiming that, even under the most favorable assumptions, the agricultural produce in Britain could at most be doubled over the next twenty-five years (cf. II.9). Over the ensuing quarter century, he thinks “it is impossible to suppose that the produce could be quadrupled.” (Cf. II.10). What he obviously means is that it cannot be doubled again, and quadrupled over the whole half century. Note that he says here that it is impossible.

All this only establishes an upper bound for the slope. Yet, Malthus then simply assumes a constant increase per quarter century (cf. II.10):

Let us then take this for our rule, though certainly far beyond the truth; and allow that, by great exertion, the whole produce of the Island might be increased every twenty-five years, by a quantity of subsistence equal to what it at present produces.

The only supporting evidence here is that “[t]he most enthusiastic speculator cannot suppose a greater increase than this.” (Cf. II.10) This sounds as if Malthus had talked this over with some expert on agricultural production. But on the whole, we have to accept all this on faith.

Malthus then goes on to present his conclusion with a drum roll (cf. II.11/12):

Yet, this ratio of increase is evidently arithmetical.

It may be fairly said, therefore, that the means of subsistence increase in an arithmetical ratio.

The claim is certainly correct if you grant the rule, but the argument is bizarre because it is no argument at all. Malthus assumes that the growth for the means of subsistence is linear, and then presents this as his conclusion, which is the fallacy of “begging the question” where you take as your assumption what you want to show. And then he himself has claimed just in the preceding paragraph that this is impossible. But now it is a general law of nature. The logic is baffling.

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His own explanation shows that Malthus does not believe that the agricultural produce of Britain could grow linearly, but at most linearly, ie. absolute increases or the slope are bounded above by a constant, not that its growth is linear. Even disregarding these logical leaps, Malthus’ argument rests on the flimsiest evidence imaginable, namely his belief that it is so. Also the historical and contemporary examples, neither in the first nor in any of the later editions, even remotely support a case for linear growth.

The most favorable interpretation that I can think of here is that Malthus means this as a challenge. As long as no one can refute his claim, it stands. But then he only makes his argument for Britain circa 1800 and for the next half century although this is supposed to be a natural law that applies always and everywhere.

To be blunt: Malthus fails to make any argument for his claim whatsoever that “subsistence increases in an arithmetical ratio.” While I am at other points uncertain about Malthus’ good or bad faith, in this case I find his treatment of one of his central tenets simply disingenuous. I have no explanation other than that he knows he has no proof.

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As I have noted above, Malthus not only fails to make his case for linear growth, he mixes the question up with the unrelated question what the slope for this linear function is. That’s why he concludes that with a population of seven million in England around 1800 at most an increase by food for another seven million is possible over a quarter century. (Note, though, that the population of England was actually 7.8 million in 1801. But the census came only after the publication of Malthus’ book in 1798, so we have to be lenient here.)

However, as already critics in his time pointed out to Malthus, this conclusion depends entirely on the reference point. When the sixth and final edition of his essay was published in 1826, Malthus makes the same argument again. But the population then stood at about 11 million people (10.4 million in 1821 and 12 million in 1831.) So his contention must now be that it was possible in 1826 to produce food for another 11 million people over the next quarter century, something Malthus had claimed in 1798 was impossible because it could at most be for seven million people. The two editions make different claims that are at odds with each other.

The funny conclusion here, as already his contemporary critics noted, is that at any point in time, a population that doubles over a quarter century is always fine for the next twenty-five years. And afterwards, you can make the same argument again, and it is fine for another quarter century, and so on. So by Malthus’ own logic, there can never be a problem with the food supply with a doubling of the population per quarter century.

There is also another hilarious conclusion from Malthus’ claim that already his critics alerted him to: If there were linear growth over time, that would also apply to the past. Now, if there were seven million people in England in 1798 and food for them, it follows that in 1773 there must have been no food at all. And before that the food supply must have been negative. All this is, of course, absurd. But then this is Malthus’ assertion that is presented as a natural law.

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As Malthus himself lets on, he thinks of his claim about linear growth for the food supply as an upper estimate. In this way, you can avoid the absurd conclusion for the past. There is an upper bound for the slope, and that does not have to imply that it was ever reached. However, then the functional form for the increase of the food supply can be anything that is in accord with this constraint. It does not have to be a linear function.

Why does Malthus then insist on this misleading wording?

The best explanation I have is because it is good rhetorics. It just sounds good to use a parallelism and say (cf. I.18):

Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio.

And if that was Malthus’ purpose, it has certainly worked. This pithy assertion has become a slogan that is supposed to encapsulate the Malthusian argument. You will find many people who write something like: “as Thomas Malthus has shown. the food supply grows linearly.” But then as you can see, he does no such thing. All he does is beg the question.

The other reason I can think of is that Malthus is still a thinker of the Enlightenment in his attitude although he criticizes it on many specific points. And such thinkers always assume that an explanation has to come in a very simple mathematical form. The paradigm here is Newton’s theory.

So while Malthus has at least a moderately plausible argument for exponential population growth — a “geometrical ratio” in his wording — he also needs an equally simple functional form for the food supply. As I have argued and will also argue in other posts, Malthus’ knowledge of mathematics was very limited even by the standards of the time. And so he perhaps just picked the only other functional form he could think of: linear growth. It was also easy to work with, another bonus.

Actually, a much more natural first guess would be quadratic growth. Think of a population on an infinite plane that starts somewhere and then expands into new lands at a fixed speed. It will settle on ever larger balls whose area grows quadratically and with it the food that can be produced on it. Of course, this cannot work for long an a sphere, only initially, but then it is at least plausible.

But then Malthus could probably not think of quadratic growth as a candidate, and so he missed this simple argument. It would have also created problems for his theory, though, because he would have had to assume that there was still plenty of uncultivated land that populations could expand into.

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One way to salvage Malthus’ claim is to interpret it as an assertion about diminishing returns. In a way, that is his point that it gets harder and harder to produce food for the next seven million. But then he never makes this argument, and it would lead somewhere else. Any function with a rate of growth (slope divided by size) that decreases would do, not only linear growth. In the sixth edition, Malthus comes close to arguing with diminishing returns when he writes in the sixth edition (cf. I.I.17):

Man is necessarily confined in room. When acre has been added to acre till all the fertile land is occupied, the yearly increase of food must depend upon the melioration of the land already in possession. This is a fund; which, from the nature of all soils, instead of increasing, must be gradually diminishing.

But then Malthus mangles the argument. The first claim here is easy to understand: There is a limited supply of land. However, the second is confusing. It seems as if Malthus thinks of diminishing returns for melioration after all land is already under cultivation. But what he writes is that the “fund,” ie. the soil, is “gradually diminishing.” This makes no sense, so I assume that he is just unable to explain diminishing returns, and means that.

Yet, in one of the subsequent paragraphs, Malthus then concedes that actually not all land is already under cultivation as is his assumption for diminishing returns from melioration as the main driver (cf. I.I.20):

Europe is by no means so fully peopled as it might be. In Europe there is the fairest chance that human industry may receive its best direction. The science of agriculture has been much studied in England and Scotland; and there is still a great portion of uncultivated land in these countries.

He distracts his readers from this embarrassing problem with a claim that is hard for them to check and implies that in China and Japan already all land is under cultivation (cf. I.I.18). But in general, he has to admit that there is plenty of uncultivated land in the world (cf. I.I.18): “There are many parts of the globe; indeed, hitherto uncultivated, and almost unoccupied; […]” Instead of addressing the problem that arises from this point, he continues with an ethical argument: “[…] but the right of exterminating, or driving into a corner where they must starve, even the inhabitants of these thinly-peopled regions, will be questioned in a moral view.” But you cannot get rid of contradictary evidence by pointing out that it is morally right that it is so.

I won’t dissect the “proof” in the sixth edition in more detail here, which despite a slightly different approach and unnecessary confusion ends in the same fallacy of begging the question. To be blunt: One of the main claims of the Malthusian argument remains unproven and there is no empirical evidence for it either. Hence this is a huge gap in the whole thesis.

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It is perhaps not all that bad, though, because you can fix this problem with a simple argument. We are on a planet with a finite surface and a finite bound for how much we can produce on an area. Hence there has to be some finite upper bound for what can ever be produced on Earth. And that is an even stronger assertion than linear growth. So Malthus could use this argument, which had been made by his precursors already.

However, he cannot do this for other reasons. If there were an upper bound for the food supply that is fixed, and population grows to the corresponding upper bound for its maximum size in no time, the conclusion has to be that the size of humankind must have been stable for a long time. Malthus knows this is not true. He even argues against David Hume that the ancient world was less populous than the world around 1800 (cf. Chapter IV).

Incidentally, Malthus was right about that. But then this is not compatible with the conclusion he would have to draw from his own theory: a stable population size for humankind since long ago. It would actually refute it. So, Malthus has to build in some growth for the food supply, but not too much, and that may be another reason why he angles for linear growth, which is not possible over the long term anyway because the food supply would have to go to infinity.

I can’t fix this problem for Malthus, and he could not do it either. So one central tenet in his argument is simply unfounded.