Calculation of Malthusian fertility and mortality from birth and death rates

Calculation of Malthusian fertility and mortality from birth and death rates

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In a simple Malthusian model, the population $p$ grows according to $p = p_0e^{rt}$, where $r = eta - mu$, $eta$ is the "fertility" parameter and $mu$ the "mortality" parameter. In the case of $eta$, the closest thing to this parameter is the Total Fertility Rate. Is there a meaningful way to convert from one to the other? An easy formula might be something along the lines of $$ eta approx frac{1}{2T} ext{TFR}, $$ where $T$ is the reproductive lifetime, but this doesn't account for gender imbalances or a more nuanced way of treating the instantaneous nature of the parameter.

If every member of the population breeds, the mean number of children born from a single individual during their whole life is $eta$ times the duration of their life. The mean life duration is $1/mu$. Hence the total fertility rate in the homogenous model you suggest is $$fracetamu$$ If only some proportion $p$ of the population can breed, the total fertility rate of each breeding individual is $$fraceta{pmu}$$

The Invisible Cliff: Abrupt Imposition of Malthusian Equilibrium in a Natural-Fertility, Agrarian Society

Analysis of a natural fertility agrarian society with a multi-variate model of population ecology isolates three distinct phases of population growth following settlement of a new habitat: (1) a sometimes lengthy copial phase of surplus food production and constant vital rates (2) a brief transition phase in which food shortages rapidly cause increased mortality and lessened fertility and (3) a Malthusian phase of indefinite length in which vital rates and quality of life are depressed, sometimes strikingly so. Copial phase duration declines with increases in the size of the founding group, maximum life expectancy and fertility it increases with habitat area and yield per hectare and, it is unaffected by the sensitivity of vital rates to hunger. Transition phase duration is unaffected by size of founding population and area of settlement it declines with yield, life expectancy, fertility and the sensitivity of vital rates to hunger. We characterize the transition phase as the Malthusian transition interval (MTI), in order to highlight how little time populations generally have to adjust. Under food-limited density dependence, the copial phase passes quickly to an equilibrium of grim Malthusian constraints, in the manner of a runner dashing over an invisible cliff. The three-phase pattern diverges from widely held intuitions based on standard Lotka-Verhulst approaches to population regulation, with implications for the analysis of socio-cultural evolution, agricultural intensification, bioarchaeological interpretation of food stress in prehistoric societies, and state-level collapse.

Citation: Puleston C, Tuljapurkar S, Winterhalder B (2014) The Invisible Cliff: Abrupt Imposition of Malthusian Equilibrium in a Natural-Fertility, Agrarian Society. PLoS ONE 9(1): e87541.

Editor: Kathleen A. O'Connor, University of Washington, United States of America

Received: July 29, 2013 Accepted: December 30, 2013 Published: January 31, 2014

Copyright: © 2014 Puleston et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work is supported by the NSF (Division of Behavioral and Cognitive Sciences) Human Systems Dynamics, Collaborative Research proposal, Development and Resilience of Complex Socioeconomic Systems: A Theoretical Model and Case Study from the Maya Lowlands [Proposal# 0827275]. Website: The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Calculation of Malthusian fertility and mortality from birth and death rates - Biology

Population growth and demographic change can be measured in many ways and ascribe to different theories that postulate changes in demographics over time and its impact on the social interactions of the population.

Demography is the study of population statistics and its changing effect on populations over time. Three of the most important components that affect the demographics are fertility, mortality, and migration.

The fertility rate of a society is a measure of the number of children born.

The mortality rate is a measure of the number of people who die.

Migration is a measure of the movement of people in and out of a country

These three components can be measured and can provide a snapshot of the changes to a population. Using these measures, theories of population growth and demographic change can be formulated to inform social interactions in populations.

There are four key theories in the study of populations growth and change:

Malthusian theory: According to Malthusian theory, three factors would control the human population that exceeded the earth’s carrying capacity, or how many people can live in a given area considering the number of available resources. Malthus identified these factors as war, famine, and disease. He termed them “positive checks” because they increase mortality rates, thus keeping the population in check. They are countered by “preventive checks,” which also control the population but by reducing fertility rates preventive checks include birth control and celibacy. Thinking practically, Malthus saw that people could produce only so much food in a given year, yet the population was increasing at an exponential rate. Eventually, he thought people would run out of food and begin to starve. They would go to war over increasingly scarce resources and reduce the population to a manageable level, and then the cycle would begin anew.

Zero population growth theory: the environment, not specifically the food supply, will play a crucial role in the continued health of the planet’s population. The theory postulates that the human population is moving rapidly toward complete environmental collapse, as privileged people use up or pollute a number of environmental resources such as water and air. This theory advocates for a goal of zero population growth (ZPG), in which the number of people entering a population through birth or immigration is equal to the number of people leaving it via death or emigration.

Cornucopian theory: Asserts that human ingenuity can resolve any environmental or social issues that develop. As an example, it points to the issue of the food supply. If we need more food, the theory contends, agricultural scientists will figure out how to grow it, as they have already been doing for centuries.

Demographic transition theory: suggests that future population growth will develop along with a predictable four-stage model. In Stage 1, birth, death, and infant mortality rates are all high, while life expectancy is short. As countries begin to industrialize, they enter Stage 2, where birthrates are higher while infant mortality and the death rates drop. Life expectancy also increases. Stage 3 occurs once a society is thoroughly industrialized birthrates decline, while life expectancy continues to increase. Death rates continue to decrease. In the final phase, Stage 4, we see the postindustrial era of society. Birth and death rates are low, people are healthier and live longer, and society enters a phase of population stability.

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.

Th e 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.

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.

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.

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.

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.

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.)

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.

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.

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.

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.”

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.

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.

Changes in U.S. Immigration Patterns and Attitudes

Projected Population in Africa

This graph shows the population growth of countries located on the African continent, many of which have high fertility rates. (Graph courtesy of USAID)

Projected Population in the United States

The United States has an intermediate fertility rate, and therefore, a comparatively moderate projected population growth. (Graph courtesy of USAID)

Projected Population in Europe

This chart shows the projected population growth of Europe for the remainder of this century. (Graph courtesy of USAID)

Worldwide patterns of migration have changed, though the United States remains the most popular destination. From 1990 to 2013, the number of migrants living in the United States increased from one in six to one in five (The Pew Research Center 2013). Overall, in 2013 the United States was home to about 46 million foreign-born people, while only about 3 million U.S. citizens lived abroad. Of foreign-born citizens emigrating to the United States, 55 percent originated in Latin America and the Caribbean (Connor, Cohn, and Gonzalez-Barrera 2013).

While there are more foreign-born people residing in the United States legally, as of 2012 about 11.7 million resided here without legal status (Passel, Cohn, and Gonzalez-Barrera 2013). Most citizens agree that our national immigration policies are in need major adjustment. Almost three-quarters of those in a recent national survey believed illegal immigrants should have a path to citizenship provided they meet other requirements, such as speaking English or paying restitution for the time they spent in the country illegally. Interestingly, 55 percent of those surveyed who identified as Hispanic think a pathway to citizenship is of secondary importance to provisions for living legally in the United States without the threat of deportation (The Pew Research Center 2013).


Scholars understand demography through various analyses. Malthusian, zero population growth, cornucopian theory, and demographic transition theories all help sociologists study demography. The earth’s human population is growing quickly, especially in peripheral countries. Factors that impact population include birthrates, mortality rates, and migration, including immigration and emigration. There are numerous potential outcomes of the growing population, and sociological perspectives vary on the potential effect of these increased numbers. The growth will pressure the already taxed planet and its natural resources.

Short Answer

  1. Given what we know about population growth, what do you think of China’s policy that limits the number of children a family can have? Do you agree with it? Why, or why not? What other ways might a country of over 1.3 billion people manage its population?
  2. Describe the effect of immigration or emigration on your life or in a community you have seen. What are the positive effects? What are the negative effects?
  3. What responsibility does the United States have toward underage asylum-seekers?


Calculation of Malthusian fertility and mortality from birth and death rates - Biology

Understanding potential patterns in future population levels is crucial for anticipating and planning for changing age structures, resource and health-care needs, and environmental and economic landscapes. Future fertility patterns are a key input to estimation of future population size, but they are surrounded by substantial uncertainty and diverging methodologies of estimation and forecasting, leading to important differences in global population projections. Changing population size and age structure might have profound economic, social, and geopolitical impacts in many countries. In this study, we developed novel methods for forecasting mortality, fertility, migration, and population. We also assessed potential economic and geopolitical effects of future demographic shifts.


We modelled future population in reference and alternative scenarios as a function of fertility, migration, and mortality rates. We developed statistical models for completed cohort fertility at age 50 years (CCF50). Completed cohort fertility is much more stable over time than the period measure of the total fertility rate (TFR). We modelled CCF50 as a time-series random walk function of educational attainment and contraceptive met need. Age-specific fertility rates were modelled as a function of CCF50 and covariates. We modelled age-specific mortality to 2100 using underlying mortality, a risk factor scalar, and an autoregressive integrated moving average (ARIMA) model. Net migration was modelled as a function of the Socio-demographic Index, crude population growth rate, and deaths from war and natural disasters and use of an ARIMA model. The model framework was used to develop a reference scenario and alternative scenarios based on the pace of change in educational attainment and contraceptive met need. We estimated the size of gross domestic product for each country and territory in the reference scenario. Forecast uncertainty intervals (UIs) incorporated uncertainty propagated from past data inputs, model estimation, and forecast data distributions.


The global TFR in the reference scenario was forecasted to be 1·66 (95% UI 1·33–2·08) in 2100. In the reference scenario, the global population was projected to peak in 2064 at 9·73 billion (8·84–10·9) people and decline to 8·79 billion (6·83–11·8) in 2100. The reference projections for the five largest countries in 2100 were India (1·09 billion [0·72–1·71], Nigeria (791 million [594–1056]), China (732 million [456–1499]), the USA (336 million [248–456]), and Pakistan (248 million [151–427]). Findings also suggest a shifting age structure in many parts of the world, with 2·37 billion (1·91–2·87) individuals older than 65 years and 1·70 billion (1·11–2·81) individuals younger than 20 years, forecasted globally in 2100. By 2050, 151 countries were forecasted to have a TFR lower than the replacement level (TFR <2·1), and 183 were forecasted to have a TFR lower than replacement by 2100. 23 countries in the reference scenario, including Japan, Thailand, and Spain, were forecasted to have population declines greater than 50% from 2017 to 2100 China's population was forecasted to decline by 48·0% (−6·1 to 68·4). China was forecasted to become the largest economy by 2035 but in the reference scenario, the USA was forecasted to once again become the largest economy in 2098. Our alternative scenarios suggest that meeting the Sustainable Development Goals targets for education and contraceptive met need would result in a global population of 6·29 billion (4·82–8·73) in 2100 and a population of 6·88 billion (5·27–9·51) when assuming 99th percentile rates of change in these drivers.


Our findings suggest that continued trends in female educational attainment and access to contraception will hasten declines in fertility and slow population growth. A sustained TFR lower than the replacement level in many countries, including China and India, would have economic, social, environmental, and geopolitical consequences. Policy options to adapt to continued low fertility, while sustaining and enhancing female reproductive health, will be crucial in the years to come.


Since Malthus made his dire warning, the human population has grown from just under a billion to more than 7 billion people. Has the human population already reached or surpassed its carrying capacity? Is the planet overpopulated with people? Do we have a human overpopulation problem?

Human Carrying Capacity

Attempts to calculate the carrying capacity for the global human population have produced widely varying estimates. However, a meta-analysis of 69 such studies concluded that the best estimate of the human carrying capacity is 7.7 billion people. The human population is projected to reach 10.88 billion people by the year 2100. If these estimates are correct, it suggests that the human population is at the tipping point and must stop growing. Some human populations already suffer shortages of food, water, and other resources and our use and acquisition of resources have already damaged the environment. Such evidence suggests that we have reached our carrying capacity and there really is an overpopulation problem.

Not Just Overpopulation

Although many environmental problems are aggravated by the size of the human population, some experts think that over-consumption and waste by populations in wealthy nations are worse problems than sheer human population numbers. People in the more-developed nations use resources at a rate more than 30 times greater than the rate in less-developed nations, where the majority of people live today. If everyone used resources at the rate of people in developed countries, the total human population would need more than one planet Earth to supply their demands. Reducing profligate consumption of resources and our ecological footprint are clearly needed to help solve the overpopulation problem.

Slowing Human Population Growth

Environmental problems are not caused solely by human overpopulation. However, having so many people on the planet certainly makes problems worse, so it is important to reduce the rate of human population growth. A widely accepted goal is an overall zero growth rate for the human population. Zero population growth (ZPG) occurs when the birth rate equals the death rate (assuming no net migration for the human population as a whole). ZPG can be achieved if women average only enough children to replace themselves and their partners in the population. This is called the replacement fertility rate. It ranges from just over 2 to almost 3 children per woman, depending on the death rate. At a higher death rate, the replacement fertility rate is higher because fewer children survive to adulthood to replace their parents in the population. Even if the fertility rate falls to the replacement level, however, there will still be a time lag before the population growth rate levels off. That&rsquos because populations that have recently had high birth rates have a youthful age distribution, with a large proportion of women at peak childbearing ages. With so many young women, the population birth rate will remain high for at least another generation.

Figure (PageIndex<4>): China&rsquos population, 1961-2015

Childbearing is a deeply important and personal decision that is influenced by many socioeconomic and cultural factors. Obviously, trying to influence how many children women have is a complex problem. A top-down approach was instituted in China in 1979 when it enacted a one-child-per-woman policy. The Chinese government has credited the policy with reducing China&rsquos population by 400 million people. However, China&rsquos fertility rate was already falling when the policy was put into effect, so the impact of the policy is disputed. The actual growth of the Chinese population is shown in the graph in Figure (PageIndex<4>). The tiny dip in the curve starting around 1979 suggests the policy&rsquos impact on population growth was minimal.

Unlike in China, most countries do not have direct policies to limit fertility rates. However, evidence from many populations shows that women start having fewer children when there are more educational and economic opportunities for females, advances in gender equality, greater knowledge of family planning, and better access to contraception. Needless to say, such society-wide changes are often very difficult to achieve and require multiple approaches, but the future of our species may depend on them.

The Theory of Demographic Transition (With Criticisms)

The theory of demographic transition or of population stages or of population cycle has many versions. It has been propounded by W.S. Thomson and F.W. Notestein. They explain the theory in three stages.

But the two famous versions are C.P. Blacker’s five stages of population growth which have been explained here, and Karl Sax’s four stages of population growth, namely, High Stationary, Early Explosive Increase, Late Explosive Increase, and Low Stationary. He does not explain Blacker’s Declining Stage, while his four stages almost resemble Blacker’s other stages.

Explanation of the Theory of Demographic Transition:

The theory of Demographic Transition explains the effects of changes in birth rate and death rate on the growth rate of population. According to E.G. Dolan, “Demographic transition refers to a population cycle that begins with a fall in the death rate, continues with a phase of rapid population growth and concludes with a decline in the birth rate.”

The theory of demographic transition is based on the actual population trends of advanced countries of the world. This theory states that every country passes through different stages of population development.

According to C.P. Blacker, they are:

(i) the high stationary phase marked by high fertility and mortality rates

(ii) the early expanding phase marked by high fertility and high but declining mortality

(iii) the late expanding phase with declining fertility but with mortality declining more rapidly

(iv) the low stationary phase with low fertility balanced by equally low mortality and

(v) the declining phase with low mortality, lower fertility and an excess of deaths over births. These stages are explained in the Fig. 1 (A) & (B) In the figure, the time for different stages is taken on the horizontal axis and annual birth and death rates on the vertical axis. The curves BR and DR relate to birth rate and death rate respectively. P is the population curve in the lower portion of the figure.

In this stage the country is backward and is characterised by high birth and death rates with the result that the growth rate of population is low. People mostly live in rural areas and their main occupation is agriculture which is in a state of backwardness. There are a few simple, light and small consumer goods industries.

The tertiary sector consisting of transport, commerce, banking and insurance is underdeveloped. All these factors are responsible for low incomes and poverty of the masses. Large family is regarded as a necessity to augment the low family income. Children are an asset to the society and parents. The existence of the joint family system provides employment to all children in keeping with their ages.

More children in a family are also regarded as an insurance against old age by the parents. People being illiterate, ignorant, superstitious and fatalists are averse to any method of birth control. Children are regarded as God-given and pre-ordained.

All these economic and social factors are responsible for a high birth rate in the country. Along with high birth rate the death rate is also high due to non-nutritional food with a low caloric value, lack of medical facilities and the lack of any sense of cleanliness.

People live in dirty and unhealthy surroundings in ill ventilated small houses. As a result, they are disease-ridden and the absence of proper medical care results in large deaths. The mortality rate is the highest among the children and the next among women of child-bearing age. Thus the birth rates and death rates remain approximately equal over time so that a static equilibrium with zero population growth prevails.

According to Blacker, this stage continued in Western Europe approximately up to 1840 and in India and China till 1900. This is illustrated in Fig. 1 (A) by the time period HS- “High Stationary” stage and by the horizontal portion of the P (population) curve in the lower portion of the figure.

In the second stage, the economy enters the phase of economic growth. Agricultural and industrial productivity increases, and means of transport develop. There is greater mobility of labour. Education expands. Incomes increase. People get more and better quality food products, medical and health facilities are expanded.

Modern drugs are used by the people. All these factors bring down the death rate. But the birth rate is almost stable. People do not have any inclination to reduce the birth of children because with economic growth employment opportunities increase and children are able to add more to the family income.

With improvements in the standard of living and the dietary habits of the people, the life expectancy also increases. People do not make any effort to control the size of family because of the presence of religious dogmas and social taboos towards family planning.

Of all the factors in economic growth it is difficult to break with the past social institutions, customs and beliefs. As a result of these factors, the birth rate remains at the previous high level. With the decline in the death rate and no change in the birth rate, population increases at a rapid rate. This leads to Population Explosion.

This is an “Early Expanding” (EE) stage in population development when the population growth curve is rising from A to B as shown in Fig. 1(B), with the decline in death rate and no change in birth rate, as shown in the upper portion of the figure. According to Blacker, 40% of the world population was in this stage up to 1930. Many countries of Africa are still in this stage.

Third Stage:

In this stage, birth rate starts declining accompanied by death rates declining rapidly. With better medical facilities, the survival rate of children increases. People are not willing to support large families. The country is burdened with the growing population. People adopt the use of contraceptives so as to limit families.

Birth rates decline a initially in urban areas, according to Notestein. With death rates declining rapidly, the population grows at a diminishing rate. This is the “Late Expanding” stage as shown by LE in Fig. (A) and BC in Fig. (B). According to Blacker, 20% of the world population was in this stage in 1930.

Fourth Stage:

In this stage, the fertility rate declines and tends to equal the death rate so that the growth rate of population is stationary. As growth gains momentum and people’s level of income increases, their standard of living rises. The leading growth sectors expand and lead to an expansion in output in other sectors through technical transformations.

Education expands and permeates the entire society. People discard old customs, dogmas and beliefs, develop individualistic spirit and break with the joint family. Men and women prefer to marry late. People readily adopt family planning devices. They prefer to go in for a baby car rather than a baby.

Moreover, increased specialisation following rising income levels and the consequent social and economic mobility make it costly and inconvenient to rear a large number of children. All this tends of reduce the birth at further which along with an already low death rate brings a decline in the growth rate of population.

The advanced countries of the world are passing through this “Lower Stationary” (LS) stage of population development, as shown in Fig (A) and CD in Fig. (B). Population growth is curtailed and there is zero population growth.

Fifth Stage:

In this stage, death rates exceed birth rates and the population growth declines. This is shown as D in Fig. (A) and the portion DP in Fig. (B). A continuing decline in birth rates when it is not possible to lower death rates further in the advanced countries leads to a “declining” stage of population.

The existence of this stage in any developed country is a matter of speculation, according to Blacker. However, France appears to approach this stage.

Criticisms of the Theory of Demographic Transition:

Despite its usefulness as a theory describing demographic transition in Western Countries, it has been criticised on the following grounds:

1. Sequences of Stages not Uniform:

Critics point out that the sequences of the demographic stages have not been uniform. For instance, in some East and South European countries, and in Spain in particular, the fertility rates declined even when mortality rates were high. But in America, the growth rate of population was higher than in the second and third stage of demographic transition.

2. Birth Rate not declined initially in Urban Areas:

Nolestein’s assertion that the birth rate declined initially among urban population in Europe has not been supported by empirical evidence. Countries like Sweden and France with predominantly rural populations experienced decline in birth rates to the same extent as countries like Great Britain with predominantly urban populations.

3. Explanations of Birth Rate decline Vary

The theory fails to give the fundamental explanations of decline in birth rates in Western countries. In fact, the causes of decline in birth rate are so diverse that they differ from country to country.

Thus the theory of demographic transition is a generalisation and not a theory.

Not only this, this theory is equally applicable to the developing countries of the world. Very backward countries in some of the African states are still in the first stage whereas the other developing countries are either in the second or in the third stage. India has entered the third stage where the death rate is declining faster than the birth rate due to better medical facilities and family welfare measures of the government.

But the birth rate is declining very slowly with the result that the country is experiencing population explosion. It is on the basis of this theory that economists have developed economic- demographic models so that developing countries should enter the fourth stage.

One such model is the Coale-Hoover model for India which has also been extended to other developing countries. Thus this theory has universal applicability, despite the fact that it has been propounded on the basis of the experiences of the European countries.

Conclusion to the Theory of Demographic Transition:

The theory of demographic transition is the most acceptable theory of population growth. It does not lay emphasis on food supply like the Malthusian theory, nor does it develop a pessimistic outlook towards population growth. It is also superior to the optimum theory which lays an exclusive emphasis on the increase in per capita income for the growth of population and neglects the other factors which influence it.

The biological theories are also one-sided because they study the problem of population growth simply from the biological angle. Thus the demographic transition theory is superior to all the theories of population because it is based on the actual population growth trends of the developed countries of Europe. Almost all the European countries have passed through the first three stages of this theory and are now in the fourth stage.

Short-run fluctuations in fertility and mortality in pre-industrial Sweden ☆

This paper describes and interprets annual Swedish data from 1750 to 1869 on weather, harvests, real wages, birth rates, and death rates using vector autogression. Impulses due to unexplained increases in wealth, whether this occurred through increased real wages, improved agricultural yields, or warmer winters, led in the short run to increased fertility and decreased infant and non-infant mortality, and hence to increased rates of population growth. Unexplained or unanticipated fluctuations in infant mortality led to replacement cycles in fertility within one to three years, although only a negligible cumulative effect on fertility persisted after five to ten years. Fluctuations in deaths among persons older than one year evoked a fertility response several years later, but this replacement response persisted after more than a decade. Although vector autoregression is not designed to account for long-term trends and their consequences, the interrelationships found here among exogenous weather shocks and fluctuations in economic conditions and demographic rates provide support for the homeostatic mechanisms hypothesized by classical economists and discussed by Malthus. The methodology of vector autoregression appears useful for studying historical series on climatic, economic and demographic variables where we do not yet have a sufficient theoretical foundation for specifying and estimating structural models.

A preliminary draft was presented at a Conference on British Demographic History, Asilomar, California, March 7–9, 1982. We have benefited from comments on the earlier draft from participants at the conference and Jonathan Eaton, Wallace Huffman, Ronald Lee, Christopher Sims, David Weir and an anonymous referee. We acknowledge the able research assistance of Thomas Frenkel and Paul McGuire and the financial support of a grant from the Hewlett foundation to Yale's Economic Demography Program.

Natural Law Theories and Social Theories of Population (With Criticism)

The thinking on population-resource nexus can be traced back to the time of Plato. However, Thomas Malthus was the first person to systematically study the subject and express many views on it.

Since his time, scholars have been eager to arrive at laws governing population growth.

The theories on population that sprang up can be divided into natural law-based population theories and social theories of population. Malthus, Michael Thomas Sadler, Thomas Doubleday and Herbert Spencer propounded natural law theories in the nineteenth century. Henry George, Arsene Dumont, David Ricardo, Karl Marx and Engels were among the social theorists. What follows is a discussion of the ideas of these theorists on population growth.

A. Natural Law Theories:

1. Thomas Robert Malthus (1766-1834):

A British professor of history and economics was the first scholar to propound a theory on population based on natural law. He examined the close link between growth of population and other demographic changes and socio-economic changes. He was keen to understand how population growth was likely to impact upon human welfare. His empirical approach to the population problem was a result of his study of the experience of West European countries.

Malthus population principle had two postulates:

(i) Food is vital for the existence of humans and (ii) the passion between the sexes is not only necessary but will also remain in the present stage. But he stressed that the power of population to reproduce is much greater than the earth’s power to provide subsistence for humans. The widening gap between population and subsistence will result in humans using the means of subsistence each for his own use.

The society gets divided into the rich have and the have-nots and a capitalistic set-up is the result. The rich who earn the modes of production earn profit and accumulate capital, increase their consumption and through this create demand for certain commodities.: The demand generates more of production. Malthus is in favour of the capitalistic set-up as distribution of the capital among the poor would mean it cannot be invested in the modes of production. So the rich will grow richer and the poor, poorer.

The widening gap between population and resources for sustaining it shall eventually lead to a scenario where misery and poverty become unavoidable. This is because of the supremacy of ‘positive’ checks, like vicious customs, luxury, pestilence, war, hunger, disease and other ills, over ‘preventive’ checks like delayed marriages and moral restraint which reduce the birth rate. Humans were doomed to inevitable suffering as they would always maintain a population greater than the available means of subsistence.

The Malthusian principle highlights the urgent need to maintain a balanced relationship between population and means of subsistence. There is a lot of truth in the Malthusian principle of population as evident in people resorting to contraception and other means to restrict their family size.

Malthus also brought the study of population into the fold of social sciences. His ideas initiated thinking that began to view dynamics of population growth in the context of man’s welfare. His principle initiated other theories by thinkers on the subject.

Some of Malthus’ works presenting his theories include An Essay on the Principle of Population, A Summary View of the Principle of Population. The Grounds for an Opinion on the Policy of Restricting the Importation of Foreign Corn and an Enquiry into the Nature and Progress of Rent.

2. Criticism of Malthus Theory:

The principle theory of population of Malthus has been critically analysed by others.

They have pointed out the following flaws:

i. Malthus’ basic premise on sexual passion has been criticised as it mixes up desire for pleasure and sex (a biological instinct) with desire for children (a social instinct).

ii. It has been noted that population has rarely grown in geometrical proportion and means of subsistence have only rarely multiplied in arithmetic progression.

iii. Malthus has assumed a time span of 25 years for a population to double itself. But the doubling period varies from country to country.

iv. Malthus established a causal relationship between positive checks in the form of natural calamities and overpopulation which is not necessary, as natural disasters can happen in under populated areas also.

v. Malthus ignored the role of changing technology and transformations in the socioeconomic set-up of a country.

vi. Malthus ignored the biological limitation of a population, that is, it cannot grow beyond a limit.

3. Michael Thomas Sadler (1780-1835):

Sadler was a British economist and social reformer. A contemporary of Thomas Malthus, he expounded the natural law of population growth as one which involved an inverse relationship between humans’ tendency to increase their population and the existing population density in an area. He propounded that all other things being equal, the population would grow only up to a point where it has secured the utmost degree of happiness for the largest possible number of humans. Sadler’s natural law of population growth theory revealed a rational basis for faith in the rapid perfectability of man’s welfare.

4. Thomas Doubleday (1790-1870):

A British economist and social reformer, Doubleday held that increase in human population was inversely related to the food supply. This meant that places where there was better food supply would show slow increase in population.

A constant population increase can be seen in places with worst food supplies, that is, the poorest people. Between the two extremes are areas with tolerably good food supplies and here the population is stationary.

It was on the basis of Doubleday’s law that Castro later (1952) said that higher protein intake reduces the fecundity of a population and low protein intake increases it. However, Thompson and Lewis (1976) have pointed out there is no scientific basis for believing that factors like population density, proportion of protein intake and the relative abundance of calorie intake have a Substantial and noticeable effect on fecundity.

5. Herbert Spencer (1820-1903):

A British philosopher, Spencer tried to explain the role of natural forces in social and biological developments. His theory of population, similar to those of Sadler and Doubleday, believed in a natural law that absolved humans from any responsibility for the control of population.

Nature would weaken humans’ interest in reproduction and make them devote more time for personal, scientific and economic development. For, the reproductive interest and capacity of individuals shall decline with their personal advancement as the latter would claim more of their time and energy. The decline in fecundity ensured a slow rate of population growth.

B. Social Theories:

The difficulty of finding conclusive evidence, about the existence of natural law controlling humans’ fecundity gave a boost to social theories on population growth.

1. Henry George (1839-1897):

An American economist and social reformer, Henry George tackled the principle of basic antagonism between the human tendency towards population increase and the human ability to provide subsistence for the increased number of people. He held the view that unlike the case of other living beings, an increase in the human population involved increase in his food as well. The threat to existence of humans is not from the ordinances of nature but from social maladjustments.

2. Arsene Dumont (1849-1902):

Stressed upon social capillarity—the tendency among people to reach higher levels socially. This tendency prevents rapid reproduction by man. Humans are then more concerned about upward mobility for personal benefit than about the welfare of the rape. This is because they believe the development of numbers in a country or society is in an inverse ratio to development of individuals. As his study was based on study of population growth in France, he pointed but that in France, the birth rates declined with the establishment of democracy and weakening of hurdles in the way of upward mobility.

In societies that are rigidly structured, mainly those in developing and under-developed countries, the social capillarity is largely inactive. However, growth of big cities in these countries has resulted in the social capillarity prevailing in the hinterlands of the cities. Here the birth rate may decline. But in areas where individual ambition does not develop much, high birth rates continue.

According to Dumont, the actual conditions under which people live are very important as the preventive checks of population growth depend upon these conditions (for instance, delayed marriages and use of contraception on an increased scale in very urbanised areas where there is greater stress on individuality).

3. David Ricardo (1772-1823):

The analytical approach of Ricardo resulted in his building up of a normative model of market system. He expanded on the labour-wage link and the impact of capital accumulation on population. Ricardo held that increased demand for labour resulted in increased wages but with increase in supply of labour, there would be a fall in wages. Ultimately, the wages shall settle at a natural wage governed by cost of subsistence.

The wage level was dependent on labour supply, which in turn was dependent on population growth, as well as capital accumulation. Population, he believed, regulated itself based on the availability of funds to employ it. So it would increase or decrease based on increase or decrease of the rate of capital accumulation.

Ultimately such an arrangement would result in a state where the demand and supply of labour shall be equated and then the capital accumulation would stop. As per the law of diminishing returns, then, there would be universal poverty as all would receive only subsistence wage. According to Ricardo, misery and poverty are inevitable in the natural circumstance.

4. Karl Heinrich Marx (1818-1883):

Marx had a distinctive approach to the population problem. He believed in the communist method of production and its ability to give full employment and good living to all workers irrespective of their rate of increase in numbers. He held that poverty and misery could be got rid off and were not a natural inevitability.

Marx held that there was no single eternal or natural law of population. Laws on population that have been put forward must be seen in the context of contemporary modes of production. Each mode of production has its own economic demographic laws.

Marx was of the view that supply of labour increases more rapidly than the demand for workers. The surplus population is then a reserve of unemployed and semi-employed people. The movement of wage levels depends on the magnitude of working population in the reserve army and the critical proportion of workers is controlled by capital contraction or expansion. The birth and death rates and family size are inversely correlated with the wage levels. A class of workers more prone to becoming a part of reserve army or surplus population shall have lower wage level and higher birth and death levels.

Marx’s views represent a theory of labour based on conditions that apply to the capitalistic mode of production. The variables his theory involves are capital accumulation, labour demand, percentage of surplus population, wage levels and demographic rates. These variables are closely linked in a system that is articulated by access to and variations in the means of employment.

Malthus versus Marx:

Comparing the theories of Malthus and Marx, it has been noted that their theories are inadequate. Both of them ignored the reality of population growing and living standards improving almost continuously over a long period. Marx argued that labour substitution would automatically result in a general fall in real wage levels, but this has not been so.

The similarities between their theories are in that (i) both were aware of the importance of labour demand as a population regulator, and (ii) they acknowledged the negative correlation between wages and birth and death rates, i.e., the rising wage level and falling birth and death rates were reversely related to each other.

“There are three fundamental differences in their theoretical formulations. One, Malthus’ conservative ideology saw self-interest as the guiding principle with marriage, family, property and inheritance. Marx was a bourgeois ideologist who held that private property ownership lay as the basic cause of most evils in the society. Two, Malthus’ principle stressed the effect of population growth and its confrontation with subsistence levels.

Population was an independent variable for him when he discussed the population- resources nexus. Marx, however, held that capitalistic production was the basic cause of the problem of surplus population and other problems it brought on. Three, Malthus’ principle has universal applicability whereas Marx’s theory is related to a particular economic system which would not operate in a feudalistic or socialist system.

The writings of both Malthus and Marx have had great significance in the study of the nature of relationship between population and socioeconomic development.

Theory of Demographic Transition:

The theory of demographic transition was put forward by W.S. Thompson (1929) and Frank W. Notestein who based their arguments on fertility and mortality trends in Europe, America and Australia.

According to this theory, when a society transforms into a literate, industrialised and predominantly urban one from an illiterate and rural-agrarian society, a particular direction of demographic change can be traced.

The theory outlines three basic hypotheses:

1. The decline in mortality rate comes before the decline in fertility rate.

2. The fertility rate actually declines to match mortality rate.

3. Socio-economic transformation of a society is commensurate with its demographic transformation.

The theory predicts conspicuous transition stages:

High and fluctuating birth and death rates and slow population growth.

High-birth rates and declining death rates and rapid population growth.

Declining birth rates and low death rates and declining rate of population growth.

Low birth and death rates and declining rate of population growth.

Birth and death rates approximately equal, which, in time, will result in zero population growth.

In the first stage, both fertility and mortality rates are high, in the range of 35 per 1000. But the mortality pattern is erratic due to prevalence of epidemics and variable food supply. This results in stable and slowly growing population.

This stage mainly occurs in agrarian societies with low or moderate population density, societies where the productivity is low, life expectancy is low, large family size is the norm, underdeveloped agriculture is the main economic activity, low levels of urbanisation and technological development prevail and low levels of literacy are experienced.

Nearly all the countries of the world were at this stage, but now to find a country at this stage of demographic transition seems improbable, because the data on fertility and mortality in such a region would be inadequate or lacking. Also, there is little chance that such a region would have remained totally unaffected by expansion in medical facilities. For these reasons, the first stage has also been called the Pre-Industrial or Pre-Modern Stage.

The second stage is characterised by high but gradually declining fertility rates (at around 30 per 1000) and a drastically reduced mortality rate of over 15 per 1000. The expansion in health, facilities and food security reduces death rates. But, because education has not reached sufficient levels, birth rates are still high.

By the end of the second stage, fertility rates start declining gradually and mortality rates start declining sharply. The population now increases at declining ‘ rates. Most of the less developed countries of the world are passing through the explosive stage of demographic transition. These countries include India, Pakistan, Bangladesh, Nepal and Indonesia.

In the final stage, both death rates and birth rates decline appreciably. As a result, population is either stable or growing slowly. At this stage, the population has become highly industrialised and urbanised technological development is satisfactory and there are deliberate attempts at restricting the family size. High literacy rates prevail. This stage is evident in Anglo-America, west Europe, Australia, New Zealand, Japan etc.


Loschky and Wildcose have criticised the theory, arguing that the theory is neither productive, nor are its stages sequential and definite. Also, the role of man’s technical innovations should not be underrated, particularly in the field of medicine which can arrest the rate of mortality.

The theory, despite its shortcomings, does provide a generalised macro-level framework within which different situational contexts can be placed in order to comprehend the demographic processes in that particular country. Also, scope should be left to take into account the fact that the present conditions are different from those that prevailed at about the end of the nineteenth century in Europe.