Malthus and His Geometrical and Arithmetical Ratios

As I have written before, I am currently working on a book: What’s Wrong with the Malthusian Argument? I will blog about some points here. But, of course, a thorough analysis goes beyond what I can cover in a few short articles.

For most people, the takeaway from Thomas Malthus’ “An Essay on the Principle of Population” is probably this (cf. I.17/18):

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.
Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison of the second.

Especially the claim about the arithmetical and the geometrical ratios has become almost a slogan that is supposed to sum up the Malthusian argument. As for the two postulata Malthus refers to, those were posited two paragraphs before (cf. I.14):

First, That food is necessary to the existence of man.
Secondly, That the passion between the sexes is necessary and will remain nearly in its present state.

What Malthus means with the arithmetical and geometrical ratios is that a population “when unchecked” grows exponentially, and the food supply — which is what “subsistence” essentially stands for — linearly. Since he is in general unable, though, to work with quantities that evolve on a continuous time scale, he always treats them as a sequence, and that’s why he speaks of a “geometrical ratio” (a fixed percentage change over a time period) and an “arithmetical ratio” (a fixed absolute increase over a time period).

Let me note that the insinuation that linear growth of the food supply follows from the postulata cannot be true. Only the first postulatum about the necessity of food could form a basis. But all you can conclude from it is that there is an upper bound for the size of a population: There can be at most so many people with a given food supply. However, that in no way implies that the food supply itself has to grow, and even less so that it has to grow linearly. As it stands, this is a complete non-sequitur, a conclusion that does not follow from its premises.

Malthus tries to provide an independent argument later on. I will look into it in another post. Basically, he confuses the question whether the food supply grows linearly with the unrelated question what the slope of this linear function is if he has already shown that the food supply grows linearly. The “proof” boils down to “begging the question” (assuming what you want to show): Let me define that the food supply grows linearly, and then, lo and behold, it grows linearly!

The situation is slightly better for the claim about exponential population growth. However, that hinges on a suitable interpretation of what Malthus means with the “passion between the sexes” and its “state.” This is not easy because he treats these terms as self-evident, and it takes some work to reconstruct a probable reading from remarks strewn over his essay.

What is even more important, though, is the meaning of the qualification “when unchecked.” Malthus is again anything but clear about his definition, and so more detective work is needed. A casual reader may also miss this qualification altogether which turns out to be extremely restrictive. As Malthus remarks later on, “in no state that we have yet known” an actual population has ever been “unchecked” (cf. II.3). Still, he tacitly drops this restriction afterwards and treats it as if any actual population grows exponentially, at least as long it has not yet hit the upper bound dictated by the food supply.

Unlike in the case of linear growth for the food supply, there is a moderately plausible argument in the background that Malthus could make although he never does so explicitly. Instead he again confuses the question of whether a population “when unchecked” grows exponentially with the unrelated question of what its rate of growth is (the ratio of the slope to the size of the population). But that presupposes that a population “when unchecked” indeed grows exponentially, which is just the question at stake that would have to be answered first.

Malthus’ implicit definition of a state of society “when unchecked” is that the population can grow at the maximum rate possible. This is so because he assumes that any actual population can have at most growth as high as an “unchecked” population. However, all this does not imply that the population “when unchecked” really grows exponentially and even at the maximum rate possible. It can do that, but that need not be the case.

As it seems Malthus wants to conclude the necessity from the possibility via the second postulatum. He also needs further assumptions that a population “when unchecked” is constrained in its behavior, which is actually contrary to the ordinary meaning of the word “unchecked.” That’s why I insist on writing the term in scare quotes. All in all, the hidden argument has a certain stringency, but mostly relies on plugging in what then drops out. I will explore this in further posts.

However, Malthus does not supply it. As noted above, he focuses on the rate of growth and not on the claim that a population “when unchecked” grows exponentially. Since he is after the maximum rate possible, any rate of growth for an actual population can serve as a lower bound. He, therefore, goes through various concrete examples and derives a lower bound that corresponds to growth that doubles the size of a population every fifteen years (and maybe even every ten years in the later editions of his essay).

Instead of taking this as his conclusion, namely that the maximum rate is bounded below by some constant, he now skips to this (cf. II.7):

[…] we will take as our rule; and say,
That population, when unchecked, goes on doubling itself every twenty-five years or increases in a geometrical ratio.

In his derivation, Malthus implicitly assumes growth at the maximum rate possible “when unchecked.” But now he contradicts this with the definition that population growth is at a rate that, according to his own argument, is clearly below it.

And then the crucial point is smuggled in at the end of the sentence, namely that a population “when unchecked” indeed grows exponentially. That follows from the assertion that it regularly doubles every twenty-five years. However, that “begs the question” again because this is just his definition!

Malthus’ main point, though, was not linear growth of the food supply or exponential population growth, but 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.

I will dissect this claim in my next post on the Malthusian argument. But let me already note that the conclusion cannot follow from the postulata as Malthus insinuates. It is a non-sequitur as it stands because the assertion about the relative growth behavior of an exponential and a linear function has nothing to do with the necessity of food for human beings or the “state” of the “passion between the sexes.” This is a purely mathematical question that only depends on the functions and not what they stand for.

To make things worse: Although the claim about the different “powers” of an exponential and a linear function is given prominence here, it is completely inessential for Malthus’ theory. His further conclusions do not hinge in any way on the asymptotic behavior of an exponential and a linear function, ie. as time goes to infinity, but on the infinitesimal situation when population size hits the upper bound. This is a red herring so big, it might be called a “red whale.” However, it has distracted many readers from the real question Malthus has to answer, namely whether the food supply grows linearly and population exponentially, which is introduced into the argument as almost a truism.

So much in this post, stay tuned …

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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?