A Problem of Prophesy

TFR vs CFR

An interesting article in The Economist today argues that current worries over falling birth rates may be mistaken. The argument hinges on a problem with predicting the future.

The critical question is how many children a woman will have in her lifetime, the completed fertility rate, CFR. If the number is more than two, population will increase, if less than two decrease. TFR, total fertility rate, is an estimate of CFR produced on the assumption that in each year of her life the average woman will have the same fertility rate, the same probability of having a child, as current women of that age, that when she is thirty her fertility will be what the fertility of thirty-year old women now is. TFR takes a snapshot of reproductive behavior at the moment, of the fertility rate of women at all ages, and predicts the behavior of current women on the assumption that the pattern will not change.

Suppose that assumption is wrong. Suppose woman are having their children at older ages than before, as in fact they are. For an exaggerated version of the actual pattern, assume that women in their twenties plan to have two children in their thirties and are not having any now, women in their thirties had two children in their twenties and are not having any more. TFR is calculated on the assumption that a woman now twenty will act as women now in their twenties are acting, have no children in her twenties, as women now in their thirties are acting, have no children in her thirties, hence that she will have no children: TFR=0. Further suppose that all of the women have as many children as they planned. Women now in their twenties and women now in their thirties all end up with two children. CFR=1 for both groups.

The point of this simplified example is that if the age of child-bearing is going up, TFR will underestimate CFR. We cannot observe CFR for women now in their fertile years but we can observe it for women older than that, so we can observe how TFR and CFR were related in the past. The answer, for American women:

In 2000 by age 26 the average American woman had produced one child; the average 32-year-old, 1.6 children; the average 40-year-old, 1.9. In 2024 the average 26-year-old had 0.6 children and the average 32-year-old 1.2. But the average 40-year-old has still had 1.9, having delayed in her 20s and caught up in her 30s. (Watch who you’re calling childless, The Economist)

The logic of the article is correct but, as the caveat to advertisements sometimes tells us, past performance is not indicative of future results and the article omits one crucial fact. A woman who decides to have her children in her thirties instead of her twenties can do it. A woman who decides to have her children in her forties instead of her thirties probably can’t. A woman who decides to have children in her fifties almost certainly cannot. The pattern of childbirth shifting to higher ages faces a biological limit.¹

I said “almost certainly.” A woman of fifty could have had eggs frozen when she was twenty and use them, possibly with the assistance of a host mother, to produce children in her fifties. The biological limit has changed, might continue to change, with continued progress in reproductive technology. With artificial wombs and the ability to convert skin cells to eggs, someone of any age and sex could produce offspring.

Life Expectancy

The issue of TFR vs CFR did not occur to me until I read the Economist article but I had noticed the same issue in a different context. During Covid there were news articles claiming that life expectancy, which has been gradually rising for many years, was now going down. Life expectancy, like TFR, is a prophecy, calculated on the assumption that what people born today will do in each year thereafter is what people of that age are doing now, new lives coming into being for TFR, existing lives ending for life expectancy. Covid resulted in substantial increases in mortality for older people. The calculation of life expectancy assumed that, since someone of eighty was now more likely to die than someone of eighty was a year earlier, someone born now would be more likely to die at eighty than someone born a year earlier and similarly for all other ages. The implicit assumption was that the pattern of mortality created by the pandemic was now permanent.

Which made for good clickbait.

Two Senses of Life Expectancy

(Addition to the post)

Life Expectancy at Birth: North America (Statbase web page)

The bar on the graph above for 1950 does not show how long, on average, someone born in 1950 will live, could not because we don’t know; many of the people born in 1950 are still alive. It is a prediction of how long someone born in 1950 will live, based on mortality rates by age in 1950. It is the equivalent for life expectancy of TFR for fertility. Call it ELE for Estimated Life Expectancy.

We cannot calculate the equivalent of CFR for 1950 but we could for 1900 if the relevant data went back that far. Almost everyone born that year in North America is dead.² If we knew at what age each person born that year died we could calculate their average as the completed life expectancy, CLE, for someone born in 1900.

We cannot calculate CLE for someone born in 1950 but we can be pretty sure that it is going to be larger than the figure on the graph, assuming no mass catastrophes in the next few decades. Life expectancy went up almost every year from then to now, which means mortality rates were falling. When someone born in 1950 reached ten, the mortality rate for ten-year-olds was almost certainly less than the mortality rate for ten-year-olds was in 1950, and similarly for every age up to the present. If we recalculated life expectancy for someone born in 1950 using actual age specific mortality rates in the years since we would have a better, and higher, estimate of CLE for someone born that year. It follows, assuming that the trend towards lower age specific mortality rates continues, that all figures on graphs like that shown are too low, that ELE is consistently lower than CLE.

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1. Source for graph below.

2. There are seven known exceptions in the US, may be a few more in the rest of North America.