An Explanation of the Malthusian Argument and Its Main Problems

[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?”]

Since it is a little difficult to keep track of the Malthusian argument in its literary form, I will try to explain it with a series of graphs. Before I can start, I need to develop a few concepts. But bear with me, some of it is a little subtle, however, not really difficult.

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The first observation is very simple: If you have a population and there is no immigration or emigration, then there are only two ways how the size of the population can change: There can be more people because children are born, and there can be fewer people because people die. That’s it, these two possibilities are exhaustive.

The point that the size of a population, barring migration, can only change via births and deaths is important, but also quite trivial once you think about it. Thomas Malthus treats it sometimes as if this were a conclusion from his theory and a deep insight, eg. when he stresses that slower population growth can only result from fewer births, the “preventive check,” or more deaths, the “positive checks,” or a mix of the two. But this is independent from his theory.

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The first idea to measure what is going on with regard to births and deaths is to look at how many of them occur over a year. Absolute numbers are not very informative, though, because they depend on the size of the population. The interesting part is the ratio of births and population size or deaths and population size. Taking a ratio means that the size of the population does not matter: If you have twice as many births and twice as many deaths for twice as many people that is the same thing as without the “twice.”

Usually, these two ratios are given for an arbitrary reference population of 1,000 people and are then called the birth rate and the death rate. Per 1,000 people, so and so many children are born and so and so many people die over some period of time, usually a year. The birth rate minus the death rate tells you how much the population grows or shrinks. If the birth rate equals the death rate, the size of the population is stable, the net change is zero.

Thomas Malthus and many of his disciples, eg. Gregory Clark, like to argue with birth rates and death rates. I think there are several reasons for this. The concept is simple, the rates are also easy to calculate from census data, and they immediately tell you something about how the size of a population evolves at the moment. However, there are also serious drawbacks with this approach.

Take the following example: At one stroke, all the older people beyond fertile age are wiped out in a country, eg. by an epidemic disease. Don’t think too much about the horror that would mean, I make this example up just to explain a point. With so many deaths, the size of the population shrinks considerably, maybe by a third. Now, assume that those who survive still have children as without the catastrophe and they also die as would have been the case otherwise. In a sense, nothing has changed for them.

Yet you observe large movements for both the birth rate and death rate. This is so because the same number of births is divided by a smaller population size, and hence it goes up. There are perhaps only two thirds as many people now as a reference population, and that means the same number of births works out to a birth rate that is 50% higher than before. This looks like people have decided to have more children, but by assumption, that has not happened. And the death rate changes, too. It goes down because the population is now younger on average than otherwise. Again, it may seem as if people were less prone to die and somehow healthier. However, nothing has changed here either, which is by assumption.

You can find many examples where authors draw mistaken conclusions from such changes in birth and death rates. For example, they might interpret the higher birth rate as a reaction where people now have more children after a catastrophic event to make up for the loss in population, or the lower death rate as the result of better conditions. But by assumption neither is the case.

My point here is that birth and death rates are a ratio of the number of births, resp. the number of deaths, divided by the size of the respective population. It is easy to miss that this ratio cannot only change because people adjust their behavior, but also because the reference population changes. The above example is, of course, deliberately extreme to make the point, but milder versions are harder to spot and easier to miss where the structure of the total population changes less obtrusively with somewhat more older or more younger people.

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Such problems with birth and death rates make two other concepts necessary that are related, but distinct: fertility and mortality. (Beware: Some authors call birth rates “fertility” and death reates “mortality,” which can be very confusing.)

The basic idea here is to look at how probable it is that people have children and how probable it is that they die. This is from the perspective of the individuals and hence independent of the population structure. In the above example, those who survive were equally likely to have children and to die with or without the catastrophe: so there is no change for fertility and mortality after the catastrophe, although birth and death rates move around.

What makes the concept more awkward than birth and death rates is that fertility or mortality is not a single number that you can easily calculate and manipulate. Basically what you need is the probability for someone of age 0 to die over the next year, of age 1, of age 2, and so forth. Mortality is a whole function that can and does vary with age. The same goes for fertility: you have a probability for someone of age 20 to have a child over the year, of age 21, of age 22, etc. Again, this is a function for all ages, not a single number (for many ages it is zero, though).

Another reason why these functions are not easy to measure is that for individuals the result might occur or not over the year. You cannot directly observe the probabilities. Only if you aggregate what happens over a larger group can you get a handle on fertility and mortality by looking at averages for different cohorts (those of age 20, age 21, etc.).

Yet another problem is that much of what you need to know has not happened yet and will only come in the future: You don’t know how probable it is for someone of age 20 now to have a child over a year five years or ten years ahead. A common simplification is to assume that the probabilities are stable over time and hence the same as for people now who are 25 or 30 years old. That makes it possible to measure fertility and mortality in one year. But this simplification depends on the assumption that the probabilities do not change over time. In case they do, and that happens often, you may introduce some errors into your calculations.

Still, thinking about demographic developments in terms of fertility and mortality is much more informative than in terms of birth and death rates. The reason is that you have a better resolution and can isolate the population structure from what happens on the individual level.

If you know the population structure (how many are in the cohorts) and fertility, you can calculate the birth rate. There are so and so many people who are so and so likely to have children, and that then leads to so and so many children. Likewise you can get the death rate from the population structure and mortality. Hence the two types of concepts are closely related.

But as in the above example: birth rates might change without a change in fertility, and death rates without a change in mortality if the population structure changes. The advantage is hence that we can discuss fertility and mortality separately from the structure of the population, whereas for birth and death rates they are lumped together.

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As noted above, birth rates and death rates are great to understand what happens with the size of a population over the next year. The change is just the difference between them. But over the longer run — that is over generations — fertility and mortality are more useful. You can calculate how many children there will be in the next generation because you can calculate how many are born and how many will survive to the relevant age. Since the population structure is separate from fertility and mortality, you can also track how it evolves and use that then to calculate birth and death rates.

It is, however, a little awkward to work with functions for all ages, especially if you want to visualize something. It would be easier if there were a single number here. One way to condense the information is to look at fertility over a whole life: people have so and so many children after all. Fertility is often treated as only for the woman, or sometimes only for the man, but I think it is easier to think of two people as the reference, which I do below.

For mortality, it is not as simple. Mortality over a whole life is just 100%. But if you are interested in how large the next generation will be, you can look at the share of those who are born and who do not survive to fertile age, eg. some age in the middle like 30 years. Of course, this is a simplification that casts important information out.

Since I want to draw some graphs, I will call the condensed numbers fertility and mortality from now on although strictly speaking the concept is more complicated and involves whole functions for all ages. Fertility in this sense is then fertility over a whole life per two people. And mortality is the share of those who are born with this fertility and who do not survive to fertile age for some mean age like 30 years.

Here is what you can immediately read off the values for fertility and mortality in this condensed version:

If per two people there are two children who survive to fertile age, the population remains stable. This is just that fertility times ( 100% — mortality) equals two. The next generation is then as large as the previous one, and so there is no growth: Two people are replaced by two people. That’s also why this level of fertility is called “replacement fertility.”

If more than two children who are born survive to fertile age, the population grows, if fewer than two do, the population shrinks over the longer run. You can also immediately write the percentage change down: If f is fertility and m mortality, then it is: f * (100% — m) / 2 — 100%. The first part is how many are born and who survive to fertile age. Then you divide by 2 because this is per two persons, and finally you have to subtract 100% to get the percentage change.

The level for replacement fertility depends on mortality. You need more children if more die until the next generation. In modern industrial societies, about 2.1 children are enough because only some 5% die until fertile age. That’s because 2.1 — 5%*2.1 is about 2. In preindustrial societies, mortality was much higher, though, often around 50% until fertile age. Hence replacement fertility also had to be higher, namely about 4, which gives 2 with an attrition of 50%, four are born of whom only two survive.

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After this preparation, I can now explain the Malthusian theory with a few graphs. The first one shows what Malthus thinks is the situation for animals, and as a first stab also for humans. They always have as many descendants as possible. For humans, this would be fertility of at least 8 children or more.

It is now a bit tricky to get fertility and mortality into the same graph because — unlike birth rates and death rates — they live on different scales. Fertility is the number of children per two people. But mortality is a percentage, the probability of surviving to fertile age, let’s say 30 years as a value in the middle.

Here is how I can still do it:

I have to make a decision on whether to work with the number of children per two people or the percentage who survive to fertile age. I pick the first one. Now, to plot mortality on the same scale, I need some corresponding number. And I take the number of children per two people that are born, but who do not survive to fertile age (let’s say age 30). That is just the product of fertility and mortality, and I call this attrition. It does not only change with mortality, but also with fertility, which is awkward as a stand-in for mortality. But it is a number now that lives on the same scale as fertility. You can also recover mortality if you divide attrition by fertility.

For a reason that I will explain in a moment, I also define another quantity, which I call excess fertility. It is just fertility minus two or how many more people there would be in the next generation if there were no mortality.

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We can now discuss how Malthus thinks a population “when unchecked” behaves. He claims that animals always have as many descendants as possible and that humans do the same if nothing holds them back. That means that fertility is not responsive to population size. It is always the same and at the maximum possible. As long as there is still enough food and there are no constraints, Malthus also assumes that mortality is always at a minimum, some baseline determined by human nature.

In the following graph, I plot excess fertility (ie. fertility minus two) in blue and attrition (ie. fertility times mortality) in red versus population size on the horizontal axis. I don’t add a scale because my explanation is only meant to be qualitative.

As noted above, fertility is not responsive to population size, and so is excess fertility with no constraints. So we have a blue horizontal line for excess fertility. And as for attrition, the contribution from fertility is a constant and as long as the population is not held back mortality is also a constant. That’s why the red line for attrition starts out also flat. However, as the population grows, food per capita goes down. At some point, there is simply not enough there for a larger population because humans need a minimum amount to survive. Hence mortality has to shoot up when people begin to starve to death, and so does attrition.

I can now explain the reason why I prefer to work with excess fertility here and not with fertility:

Where the two lines intersect, excess fertility and attrition are the same. That means that if N children are born with the respective fertility, (N — 2) die until fertile age (= excess fertility). But then two survive, and we have replacement fertility. So it is easy to see at what size the population stabilizes, it is where the two lines cross.

There is also another advantage here: We can immediately see what happens from one generation to the next. The formula for the percentage change was: f * (100% — m) / 2–100%, where f stands for fertility and m for mortality. We can rewrite this as (f — 2)/2 — f*m/2 = (excess fertility minus attrition) / 2. But you can get that also directly: per two persons, excess fertility is the number of children born beyond two. If you subtract the number of children born who do not survive to fertile age, ie. attrition, you obtain the number of children beyond two, ie. a stable population, who survive to fertile age. Since the reference are two people, you have to divide by two to get the percentage change.

Hence the distance between the two lines is just the percentage change over a generation up to a factor of 1/2. That’s also another way to see that there is no population growth at the intersection of the two lines because the distance is zero there. My definitions were perhaps a little obscure, but the graph is now very easy to interpret: half the distance between the lines is the percentage change from one generation to the next. (It is little trickier than that because fertility and mortality may not be at the same time, but this only meant as an approximation.)

If the population starts out on the left with a small population size, it will grow by a fixed percentage from generation to generation, which is Malthus’ claim that it has geometrical growth, or on a continuous time-scale: exponential growth. It grows in this way as long as fertility and mortality remain at their maximum and minimum, respectively, and until the population hits a wall of rising mortality (and here: attrition) because there is a constraint from the food supply. That forces it to stop growing. Mortality becomes so high that maximum fertility is replacement fertility. (The population could go beyond that size, but only for a short time because death from starvation would soon push it back down. It is the maximum size over the longer run that cannot be breached.)

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Note that such a population “when unchecked” keeps having maximum fertility even when it runs into a constraint from the food supply. Fertility is totally unresponsive to anything. I don’t think this claim is even true for animals, or at least not for all species. It is certainly false for humans. Think about what that would mean. Exponential growth with maximum fertility and even a mortality of 50% as in preindustrial societies is very fast. Per two people there would be eight children (or maybe even more). Half of them die until fertile age. But that means the population doubles every generation, two people become four people. That is also roughly what Malthus derives as a result.

Now, with a doubling per generation, it would only take about ten generations for a population to grow to 1,000 times its size, twenty generations means 1 million times its size, thirty generations 1 billion times its size. That would take only 900 years with a generation length of 30 years, less than a millennium. Humankind should have long ago hit the maximum size and would have remained there ever since. What you would hence observe all around is that people have eight children per two people of whom six die until age 30, some from baseline mortality, but most from sheer starvation.

You would hence regularly find mortality of 75% (or even more), and it would to a large extent work via starvation. However, that has never happened for any human population over the longer run. It can, of course, happen sometimes with extreme events like sudden catastrophes, but here it would always have to be so. For populations far away or distant in the past, Malthus and his readers might have been unsure whether that could not have been the case although that is also false. Yet for their own times and society they only had to look around and see that this could not be a reasonable explanation of reality.

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That’s why Malthus needed to adapt his model. He still thought it was basically correct, but tried to fix it somewhat. One way here is to fiddle with the assumption that mortality only jumps up at the last minute when food becomes too scarce. If populations are not always on the brink of starvation, obviously they have to stop growing earlier.

One way how this could work is that mortality increases already before the brink of starvation is reached. There could be other factors apart from literal starvation that lead to higher mortality. Malthus calls them the “positive checks.” In principle there are three types of factors here:

• Extra mortality unrelated to population size. Then such a factor would shift the red line upwards.
• Extra mortality directly dependent on population size. That would mean that the red line is above the baseline and changes in the horizontal direction. More population could mean more mortality in and of itself. The red line would then slope upwards over the whole range or only from some point on.
• Extra mortality dependent on the constraint for the food supply, but already before it hits. As the population grows, there would be less and less food per capita. People would perhaps not yet starve to death, but they would be undernourished, and that could lead to more mortality. In this case, the red line would slope upwards towards the right end. If the population has abundant food for small population sizes, how abundant it is might not matter. So further to the left, the effect could be zero although that need not be the case.

In principle, it is also conceivable that the red line sometimes slopes downwards, ie. that mortality goes down with more population. If smaller sizes lead to less efficient food production, a larger population might be advantageous. However, Malthus never considers this possibility. In keeping, I will ignore it from now on although it cannot be ruled out.

What we are left with are two plausible possibilities: The red line is shifted upwards as a whole because of extra mortality unrelated to population size. Or it could slope upwards from left to right, maybe over the whole range (more plausible if population density plays a role in and of itself) or more to the right (if there are indirect effects from less and less food per capita).

Here are a few mechanisms that Malthus considers and that could lead to extra mortality. Since I want to avoid a discussion of fluctuations over time and of infrequent, but recurring events, I think of the red line as an average over time. One-offs result in a setback where the population is thrown to the left and then follows the same dynamic as before.

First some factors that might be unrelated to population size:

• Wars could wipe out part of the population. If that is on a rather regular basis, it would shift the red line upwards on average over time.
• Regular epidemics could have a similar effect and shift the red line upwards if their effect is unrelated to population size.
• Natural catastrophes could also happen at any population size and shift the red line upwards.

Next some causes for extra mortality that should depend on population size, but could be independent from the food supply:

• Population size via population density for a fixed area could lead to extra mortality because of more crowded housing, closer contact between people, etc. and their effects on health. For example, contagious diseases might spread faster. Malthus has this intuition, and I think it is plausible.
• Higher population densities could also lead to more aggression and maybe even wars as Malthus can imagine. That was not a new theory, already Sir Walter Raleigh had speculated about that. I will not develop the argument here, but think that this is at least overblown, or very probably false as a general explanation for wars.

And then there are causes for extra mortality that might result from less and less, though still sufficient food per capita:

• Starvation might affect a population differentially. Although, on average, there could still be enough food, some might feel the pinch earlier on.
• A barely sufficient food supply would lead to undernourishment (perhaps not because of the level, but because of less food security that exposes people to periods of undernourishment). This has certainly a detrimental effect on the immune system. And that again could then lead to more deaths from diseases or an easier spread of epidemics.

In principle also the mechanisms that only depend on population size could interact with a strained situation for the food supply. For example, more crowded housing could make people more vulnerable to diseases who are also undernourished, and that could be worse than any of the two. Or the prospect of imminent starvation could make people even more aggressive and warlike than population size alone.

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Whatever the factors may be, and if we can exclude that the red line may sometimes slope downwards, we should have something like this:

The blue line is again constant excess fertility (fertility minus two), which is still assumed to be independent from population size. We are here in the case where humans behave as Malthus thinks animals do: they always have as many children as possible. The red line is the same as above and stands for attrition (fertility times mortality), which is at first independent from population size, but then shoots up at the maximum population size when food per capita hits a minimum that people need to survive.

The yellow line incorporates other indirect effects from the food supply and also from population size (probably via population density). I leave extra mortality independent from population size out that would shift the yellow line upwards, and also effects that lead to a slope upwards over the whole range. The reason is that these are rather unimportant and do not change the conclusions materially. I will address that below.

The yellow line starts out flat and then begins to rise already earlier on than the red line. It is above the red line because there are causes for extra mortality. However, all this does not matter a lot for the dynamics. As long as the blue line is above the yellow line, the population grows. Initially, its size increases exponentially as with the red line. But when the yellow line starts to slope up, growth slows down until the yellow line crosses the blue line. That’s where the population now hits a hard constraint. It cannot grow beyond this population size (or only for a short time), which is smaller than with the red line because of extra mortality. Yet, the general pattern is the same.

The reason I skipped a shift upwards is that it does not matter for the general behavior. It only leads to slower exponential growth initially. If the yellow line shifted so far up that it were above the blue line, then the population would be doomed because it shrinks at any population size. It would probably also not be there in the first place because the first settlers on the territory would already face a shrinking population size.

The reason I skipped the case where the yellow line slopes upwards already from the start is similar. That would only mean that growth slows down earlier. But the relevant point is only that the yellow line intersects the blue line at some point and that’s where population growth must stop because it is not possible for the population to grow any further. You can add these factors to the graph if you like, but the story remains basically the same.

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What’s the difference with the case for the red line where only starvation and famine stop population growth?

At the intersection of the yellow and the blue line, people still have maximum fertility of eight or more. The lines only intersect if all but two of the children die who are born per two persons. That means mortality is still the same as in the previous case: always 75% or more, which is contrary to what we know about actual populations.

What changes is that now this high mortality versus a baseline does not stem from starvation and famine alone. The population is in as dismal a state as in the previous case, they only die from other causes than starvation, for example:

• Causes that are independent from population size: wars and epidemics independent of food or population size as well as natural catastrophes.
• Causes that depend on population size, mostly perhaps via population density, but not the food supply: extra mortality from more crowded housing, closer contact, and more aggression.
• Causes that depend on the food supply: not only outright starvation, but also indirect effects from undernourishment and less resistance to diseases.

All these factors may now determine the maximum size of the population that is possible. Factors driven by the food supply might play a role. But the crucial point is that they might not. It is entirely conceivable that only factors that depend on population size or are independent from it, but not the food supply increase mortality enough so that the yellow line hits the blue line.

Hence in principle, it is possible that the population could stop growing before the constraint for the food supply affects it at all, whether directly or indirectly. That would be a situation like this: People have more than sufficient food, but they die from things like wars, epidemics, natural catastrophes, etc. that are independent from population size, and from things that depend on population size alone, eg. more crowded housing and its detrimental effects.

The problem for the Malthusian argument here is that it is no longer clear that a constraint from the food supply determines the eventual population size. It might, but it also might not. If it doesn’t, then the population is simply unable to grow so large that the constraint from the food supply becomes relevant. It would still not be a pleasant situation because people have the same high mortality as “when unchecked,” only for other reasons. The “positive checks” alone cannot improve the lot of the population.

Already this case contradicts Malthus’ fixation on the food supply as the only cause that can stop population growth. His exposition is so convoluted that he and probably also his readers miss this serious problem for his theory. Basically, Malthus treats the “positive checks” as a slight perturbation that leaves the results for the situation “when unchecked” intact. But in this crucial regard, that is not true. The food supply may play a role in determining the eventual size of such a population or it may not. What remains true, though, is that the population will grow to a maximum size it simply cannot go beyond. This part of the story carries over also with the “positive checks.”

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The lot of the population is as grim as without the “positive checks.” However, this is still too far away from what Malthus and his readers knew was realistic at least for their society and time. Mortality until fertile age would have to run at 75% or even more, hard to square with high, but considerably lower mortality at the time. Also people did not have maximum fertility by far. Malthus points to some examples, like the United States, where fertility perhaps approached a maximum. However, that only shows that that can sometimes happen. As for Europe, it was certainly not so.

To handle this awkward fact, Malthus concedes that humans — unlike animals in his view — can also have control over their fertility, which he calls the “preventive check.” He treats this as another perturbation to the situation “when unchecked” that still leaves his conclusions intact. But then this addition makes the model so flexible that it can explain anything, which is, of course, convenient if you have to explain contradictory evidence away.

As for how Malthus thinks a population can control its fertility, he mostly focuses on late marriage. The implicit assumption here is that married couples will always have maximum fertility. If that is true, shortening the time of fertile age within marriage would also lead to less fertility overall. In the later editions, Malthus also considers “moral restraint” and even as on a par with the “preventive” and the “positive checks” although he does not mention it in the first edition.

There are also discussions of other practices that result in lower fertility. Polygamy appears as unnatural because Malthus views monogamy as the natural choice for humans. He also condemns birth control on moral grounds where he probably refers to contraception and abortion, both known at the time. In addition, he claims that promiscuous sexual intercourse results in low fertility although his argument remains obscure.

Since the central claim of the Malthusian argument is that population growth is constrained by the food supply, Malthus endeavors to show a connection also with lower than maximum fertility. As he argues, humans — unlike animals — can foresee suffering for their children. They then react to this prospect with lower fertility. In as much as suffering would result from a dwindling food supply per capita, this would create a connection.

However, as explained above, there can also be factors that lead to extra mortality and that are unrelated to the size of the population and others that only depend on the size of the population, but not the food supply. Hence while this connection may lead to a reaction when a population approaches the maximum size possible, the reason for this may not be a constraint from the food supply or only partially so.

And there is still another problem for Malthus here that he needs to handle: Many classes in society, before all the aristocracy, were nowhere near starvation. But they still did not have maximum fertility. Actually, if they had had higher population growth than the rest of the population, their descendants should have squeezed all other classes of society out. As William Godwin cleverly asked in his reply to Malthus: Why are the English not all of noble descent?

To get around this problem, Malthus concedes that there could also be other reasons for lower fertility independent from the food supply. Those higher up in society might fear that their children fall down the social ladder if they have too many of them. Hence they would reduce their fertility, too. But that has nothing to do with the food supply. Still it is conceivable that there is a connection with population size. An aristocrat might resent dividing his estate among too many children. In as much as their is a constraint for how many large estates there can be, this would then lead to an effect from the population density of nobles. Malthus considers also similar mechanisms for other classes in society where population density, not the food supply might be the driving force.

His implicit assumption is, however, that the poor are too stupid or inconsiderate to practice restraint or that it plays a minor role with them. Only imminent misery for themselves and their children will force them to lower their fertility. As Malthus is out to prove that the food supply is the binding constraint, he focuses on the threat of starvation here. Yet, in view of the discussion above, any rise in extra mortality should lead to the same result whether it is driven by the food supply or by population size alone.

We have parallel causes here to the case for mortality that lead to less than maximum fertility and excess fertility:

• Causes that are unrelated to population size, eg. that people make the voluntary decision to have fewer children although they neither face a constraint for the food supply per capita nor an impact resulting from population size. “Moral restraint” might also fall under this heading. Or people just don’t always have enough interest in procreation to achieve maximum fertility, which may depend on other circumstances, eg. customs or religious beliefs. All this would shift the curve for excess fertility downwards.
• Causes that depend on population size, most probably mediated via population density. Those could affect all classes in society. They could be related to the food supply, but not in the sense of a binding constraint. For example, the population might lower its fertility when food becomes more expensive or when food security gets worse although there is no immediate or foreseeable effect on mortality. It is plausible that such causes would add a downward slope to the curve for excess fertility, maybe over the whole range, but maybe also only from some point on.
• Causes that result from threatening prospects for the food supply, or more generally rising mortality whether from the food supply or not. Again, this should lead to a downward slope for fertility, though probably one that only begins more to the right with high population sizes.

In the next two graphs, I try to capture two possible ways how lower fertility could work out. As in the case for mortality, I skip the case for causes that are independent from population size and just shift the curve for excess fertility downwards. I also skip causes that add a downward slope over the whole range. Both cases may occur, but do not change the conclusions materially. The argument here is parallel to the one for mortality.

The first example is with a moderate decrease for fertility. Excess fertility is shown as the purple line. Fertility is first constant, but then slopes downwards from some point on, somewhat before mortality starts to rise. That would mean that the population anticipates problems ahead.

Since I do not plot mortality itself, but attrition, the product with fertility, the yellow curve depends also on fertility. It is the same as above. However, attrition now slopes downwards with fertility although mortality remains constant. Only later mortality itself increases and with it also attrition:

In this example, the population begins to grow exponentially from a small size, visible as the fixed distance between the two lines. Then fertility decreases and somewhat later mortality starts to increase. This narrows the distance between the curves down, which means slowing growth. At the point where the two curves intersect, there is no growth and the population stabilizes. This happens earlier than it would with constant fertility at the maximum.

The eventual size of the population is now determined by the factors that lead to extra mortality and the factors that lead to less than maximum fertility. As noted above, none of the factors for extra mortality may be related to a constraint for the food supply. Only factors that depend on population size alone could suffice. And also none of the factors for lower fertility need to depend on a constraint for the food supply. They might be driven by population size alone or by factors that lead to extra mortality, but not because of a constraint for the food supply.

Hence, it is entirely possible that the population stops growing before the constraint on the food supply begins to play a role at all. Of course, direct and indirect effects from a constraint for a food supply may matter or even be the dominant driver. But Malthus’ insinuation that this is mostly or even always the case does not follow from a setup with both “preventive” and “positive checks.”

In this example, both decreasing fertility and rising mortality play a part in determining the eventual size of the population. There is some trade-off between the two: If fertility decrease less, there has to be more mortality, if it decreases more, there has to be less mortality. That is basically the trivial observation at the start that there are only two possibilities to change the size of a population: births and deaths.

But it is not a simple wash: The population is now better off than in the case with only “positive checks” or “when unchecked” because it has less excess fertility than the maximum when it stabilizes. Hence to achieve replacement fertility it needs also less attrition and, therefore, also lower mortality.

Another important observation here is that the population does no longer grow to the maximum size that is possible. It could grow larger now by raising its fertility. So there is leeway on the upside. This contradicts Malthus’ claim that the food supply must always keep the size of the population down. But here it is not kept down, only when the population raises its fertility to the maximum, which results in the previous case.

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But then, it is not even necessary that population growth stops because of a combination of rising mortality and decreasing fertility. The latter could do the trick alone. Actually, although Malthus apparently excludes this case, it is something we all know: The population size in modern industrial societies does not result from rising mortality at all, only from sufficiently low fertility. There is a lot of leeway here on the upside, populations could grow massively without any impact on mortality.

Here is an example where the reduction in fertility is more pronounced:

The yellow line is the same as in the previous examples. Mortality is constant over the range shown, and attrition, the product of fertility and mortality, slopes down only because of decreasing fertility. The green line is for excess fertility, which decreases from some population size on.

In this example, fertility decreases so early on that it hits the curve for mortality already before any effects from the food constraint or population size begin to raise mortality. Hence mortality plays no role in determining the eventual size of the population, only fertility. For given fixed mortality, the population reduces its fertility to the corresponding replacement level.

The decrease in fertility can be driven by all the factors enumerated above. Some of them might depend on a constraint from the food supply. For example, the population could sense a less abundant food supply ahead although it does not have an effect on mortality yet. This could work via rising food prices or more, but still manageable food insecurity. But the constraint on the food supply may play no role at all here as is the case in modern industrial societies. The population might just stop growing before the constraint on the food supply comes into sight, however remotely.

The remarks above also apply here: The population in this example grows exponentially at low sizes, then its growth slows down until it stabilizes at some point. The respective population size is not the maximum that is possible. The population has leeway on the upside if it raises its fertility, and maybe even massively so. Hence there is no longer inevitable “population pressure” that drives a population forward until it is stopped by rising mortality.

The food supply might not play a role in determining the eventual size of the population, viz. modern industrial societies. Hence Malthus’s sweeping claim that “[i]t is an obvious truth, which has been taken notice of by many writers, that population must always be kept down to the level of the means of subsistence” (cf. P.3) is no longer necessarily true once he concedes that there are the “preventive” and the “positive checks.” As shown above, the “positive checks” alone make this conclusion false. But with the “preventive check” also inevitable population growth until stopped by mortality is out the window.

Due to the complicated exposition in his essay, Malthus and probably also many of his readers lose track that the derivations in the case of a population “when unchecked” do no longer carry over when the model is made so flexible that it can accomodate any development. It appears as if the addition of the “checks” were only a minor perturbation that leaves conclusions intact that were derived for the case “when unchecked.” But that is false.

There is another fundamental problem here. As noted at the start: There are only two possibilities if there is no migration: Population growth can be lower because of higher mortality than a minimum (the “positive checks”) or because of lower fertility than a maximum (the “preventive check”). But then any development can be explained if you make both fertility and mortality variable. You can always call the actual fertility of a population the result of the “preventive check” and its actual mortality the result of the “positive checks.” There is nothing you cannot handle in this way, and that makes the whole theory immune to refutation.

Malthus makes ample use of this when he discusses actual populations worldwide and in history. Of course, with this much flexibility, it is always possible to account for any observation. Malthus takes this as confirmation for the wide applicability of his theory although it is based on a triviality. What it actually shows is that the theory lacks any explanatory or predictive content.

At least in the first edition, the main argument is deductive, so the empirical examples could count as illustrations. But in the later editions, Malthus purges his deductive proof and relies only on an epic discussion of examples that go from Chapter III of the First Book to Chapter X of the Second Book, stretching over twenty chapters. However, that only hides the logical problem that it is not possible to prove his theory in this way because it already breaks down in theory.

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Let me jot down the important conclusions from this discussion:

• Malthus starts with a population that has always maximum fertility and where only a constraint from the food supply can stop population growth when mortality shoots up. He draws the conclusion from this that such a population will find itself at the brink of starvation in no time and will remain there. This is the maximum population size that is feasible.
• Mortality must be much higher than is observed for actual populations, though, and it must mostly result from starvation. To handle the obvious objection that this is contrary to the evidence for actual populations, Malthus introduces the “positive checks,” which lead to extra mortality, and the “preventive check,” which leads to lower than maximum fertility. He views those as slight perturbation of the ideal case “when unchecked” and insinuates that his conclusions carry over.
• Malthus mostly focuses on “positive checks” that are indirect effects from the constraint for the food supply. He mixes this up with factors that lead to extra mortality with rising population size, but which may not be driven by the food supply. In addition, he mostly ignores factors that might be independent of both the food supply and population size.
• If we add the “positive checks,” but keep fertility unresponsive to population size and constant at the maximum, then most of the conclusions carry over. The population will grow to the maximum size possible although its growth slows down earlier. Mortality has to rise as high as without the “positive checks.” The eventual size of the population is determined by the factors that increase mortality. That can be a combination of factors that are independent from population size, dependent on population size alone or result from the constraint for the food supply. It is not necessary that the latter plays a role, and so the conclusion does not remain intact that the food supply determines the eventual population size.
• To fix unrealistically high mortality in the endgame, Malthus also introduces factors that can decrease fertility. He again focuses on those that are related to the constraint for the food supply. However, he also concedes that there can be factors that depend on population size alone, eg. for classes in society that enjoy food security, or not at all. eg. when people practice “moral restraint” or are just not as interested in procreation as he thinks.
• A population with both the “preventive check” and the “positive checks” will grow to a certain size, which is no longer the maximum size possible because there is leeway on the upside with increased fertility. So, another conclusion from the case “when unchecked” does not carry over.
• The eventual population size is determined by factors that increase mortality and decrease fertility. It may depend on factors that are driven by the constraint for the food supply, but that need not be so. Hence the conclusion that only the food supply can explain the eventual population size does not remain intact as already with the “positive checks” alone.
• The eventual population size may even result only from factors that lower fertility, and not from factors that raise mortality. These factors may include some that are driven by the constraint on the food supply, but that need not be the case. A population can hence stabilize with no regard for the food supply. Modern industrial societies are a clear example for this.
• The “preventive check” and the “positive checks” add so much flexibility to Malthus’ model that it becomes immune to refutation. Malthus can account for any development, which is convenient to explain actual examples away that do not conform to his predictions. But that means it can also accomodate population dynamics where the gripping conclusions for a population “when unchecked” do not carry over.
• A population need not grow to the maximum size, it may stabilize at any size and also at a size that is so far from a constraint for the food supply that that plays no role in determining its eventual size. There is also no longer inevitable population growth, or “population pressure,” that makes the Malthusian endgame inescapable.