Economics/demographics question: If a child unexpectedly dies, how much does this shrink the next generation?

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Aug 7, 2014

The answer seems obvious - the next generation will have one fewer person (in expectation) - but it's not that simple, and it's been bugging me for about a day now.

Suppose you are an average 15-year-old, and your parents are too old to have any more children (they won't have more children to "replace" you). The ~2 children you would have had obviously won't be born. Naïvely that means the next generation will be smaller by 2, but this disagrees with the obvious answer (smaller by 1).

Where this reasoning goes wrong is in assuming that everyone else will still have the same number of children. The sex ratio will shift so that the surviving members of your sex have n more children, and the size of the next generation will decrease by 2 minus n. If n is 1, we get the intuitive answer that there'll be 1 less person.

But there's no reason why n has to be 1 for both sexes! If both a boy and a girl die, the sex ratio is unaffected and the next generation will be 1 smaller, so n has to average to 1, but n may or may not be the same between sexes. Have there been any studies estimating the value of "n" for each sex?

(I posted this because it's relevant to population ethics, but I'm not entirely sure whether it belongs here, so I also posted it to Reddit. Should questions like this go in Discussion or in an open thread?)

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It is equivalent of turning the clock a little back in the population size time series. Assuming a constant capacity, these curves are idealized as logistic functions (although delays in the feedback mechanisms can induce overshooting).

If the population is in the stationary region of the logistic function, then there is no significant effect: the person who dies is replaced by somebody else using the freed up resources.
If the population is in the exponential region of the logistic function, the loss of a single person will reduce population size by 1 in the current generation, k in the next generation, k^2 in the second next generation and so on, where k is the (unisex) growth rate. In order to take into account gender differences, replace the first k in each of these products by a gender-specific rate.

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Certainly in an idealized world the reproductive capacity of a tribe of humans is only limited by the number of women. C.f. Randy the guinea pig, father of 400.

But on the other hand, neither modern humans nor ancestral humans lived in that kind of idealized world. In the modern world we have limited monogamy and reduced pressure to have kids. Somewhere around 18% of women in the U.S. don't end up having kids - I'd expect that a woman surviving would lead to more kids, but not actually 2 more, and similarly a missing man wouldn't just be replaced by the nearest available sperm-producer. I dunno how to put a number to it.

In an ancestral environment close to equilibrium (what you imply by saying that each person has 1 kid on average), the situation is even more egalitarian. That equilibrium is maintained by something other than birth rate. If the issue is limited resources, and if an additional person can gather additional resources, then a man and a woman will both be able to increase the long-term number of children about the same. If the population is growing exponentially but is occasionally devastated by war, a man will lead to a larger population the war is in five years but a woman will lead to a larger population if the war is thirty years. If by disease or famine, there might be very little dependence on gender.

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I'd expect that a woman surviving would lead to more kids, but not actually 2 more, and similarly a missing man wouldn't just be replaced by the nearest available sperm-producer. I dunno how to put a number to it.

One way to start estimating it would be to correlate local sex ratios with local birth rates and try to control for as many things as possible. Unfortunately, this is probably very hard to do...

In an ancestral environment close to equilibrium (what you imply by saying that each person has 1 kid on average), the situation is even more egalitarian.

I'm actually most interested in the answer for modern poor countries, which are neither stable in population nor Malthusian. Basically, I'm wondering how interventions that save lives of one gender (but not the other) today will affect the population size 20 to 30 years in the future. Non-replacement fertility doesn't qualitatively change things: the question just becomes whether a life saved increases the population by more or less than "next generation's size / current generation's size". Replacement fertility is just the special case where the ratio is 1; I used that number in my question only for simplicity.

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Long term, it depends upon what the constraints are upon population size.

For example, if it happens in an isolated village where the food supply varies from year to year due to drought, and the next year the food supply will be so short that some children will starve to death, then the premature death of one child the year before the famine will have no effect upon the number of villagers alive 20 years later.

The same dynamic applies, if a large factor in deciding whether to have a third child is whether the parents can afford to educate that child, and the cost of education depends upon the number of children competing for a limited number of school places.

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Suppose generation 0 is the parents, generation 1 is the generation that includes the unexpectedly dead child, and generation 2 is the generation after that (the children of generation 1).

If you are asking about the effect upon the size of generation 2, then it depends upon the people in generation 1 who didn't marry and have children.

Take, for example, a society where generation 1 would have contained 100 people, 50 men and 50 women, and the normal pattern would have been:

10 women don't marry
40 women do marry, and have on average 3 children each
30 men don't marry
20 men do marry, and have on average 6 children each

And the reason for this pattern is that each man who passes his warrior trial can pick and marry 2 women, and the only way for a woman to marry to be picked by a warrior.

In that situation, having only 49 women in generation 1 would make no difference to the number of children in generation 2. The only effect would be having 40 women marry, and 9 not marry.

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If you have one less child, the next generation would use one person's less worth of resources, and would therefore be able to support its population in marginally more comfort. Everyone else will have marginally more children because their children can be supported in marginally more comfort, which will wipe out most of the change from having one less child yourself.

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To clarify, I'm asking about "sex-ratio effects" (which are always important) and not "resource effects" (which only matter when reproduction is resource-limited).

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I think you should modify the post to clarify this.

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This is true only if we assume childbearing is constrained by available resources (i.e. Malthusian assumptions), and in the First World we don't seem to be close to Malthusian bounds. Under real-world conditions in this or a near-future timeframe, I'd expect the resource effects of a childhood death on subsequent generations to be complex, and probably to depend mainly on the kid's counterfactual career path and luck. I'd expect the population effects of a death to be similarly complex but to instead involve demographics, ideology, and education (both of the child and of others), along with the obvious loss of one possible parent.

(You'd actually be interested in per-capita resource use, not pure resources, so there's an efficiency term too, and administrative and service work generates complicated downstream effects. But that's not important as long as the above continues to hold.)

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This is true only if we assume childbearing is constrained by available resources

No, it's not. We only need to assume that people have more preference for a child when the child would live in a world with more resources. The resources don't have to be low enough to affect the chance of the child's survival. If someone is more likely to have a child if the child could live an upper middle class lifestyle, and one less person means there's a marginally greater chance that the child could live such a lifestyle, the effect would happen.

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"Constrained" may have been a poor choice of words, but I still don't think that's a good assumption. It's plausible that people are more inclined to produce children when they expect them to be economic assets, but outside of a fully exploited subsistence environment I don't expect that to depend linearly on population. And even if it did turn out to be linear, I don't think our instincts are accurate enough to make that determination without something extremely simple and obvious (like remaining land area for farming) to cue off of.

Note that richer countries have fewer children per capita than poor ones, and richer demographics within those countries tend to fewer children than poorer ones. There's some confounders like infant mortality to deal with, but even so this would be strongly antipredicted by your theory.

(On the other hand, I'd be willing to believe that a given set of parents is likely to want a certain number of children, and to produce more up to that number, fertility allowing, if one unexpectedly dies. Inheritance and other family-controlled resources can play into this, but it remains a purely local phenomenon.)

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Note that richer countries have fewer children per capita than poor ones, and richer demographics within those countries tend to fewer children than poorer ones.

I can think of several reasons offhand why rich people would have fewer children:

The lifestyle conducive to being rich makes the parents more likely to delay childbearing, possibly leading to having fewer children.
Rich people are smarter, more educated, and/or have better impulse control and therefore are more likely to use family planning properly.
Raising a child is disproportionately costly for richer people--they may spend lots of money on the child for things like college educations, and they don't have jobs such as farming which would let them benefit from using the kids as manual labor.

The first factor applies to the degree that becoming rich is controlled by the parents' actions. The second applies to the degree that being rich is associated with the parent's traits. And the third applies to the degree that the parents want to achieve a particular level of wealth for their children and need to spend money to do so. Having the children become marginally richer because of factors that are not related to the parents' actions or traits and do not involve spending money on them would not lead to that marginal increase being correlated with having fewer children.

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