New evidence that declining fertility rates may not stabilise

It is not news that total fertility rates across most countries are declining, and have been for some time (see the links at the end of this post). However, in spite of declining fertility, many population projections assume that fertility decline will come to an end, or even reverse. For example, the UN World Population Prospects projections assume that fertility rates eventually enter a relatively stable phase, with an average long-run level of about 1.6 births per woman, although the assumed stabilisation level varies across countries (see page 29 of this methodological report). Similarly, Stats NZ assumes that the median period total fertility rate will slowly decline from 1.59 in 2025 to 1.55 in 2054 and beyond (see here).

Given that it has been relatively rare for total fertility rates to reverse trajectory, how realistic is it that they will attain some stable equilibrium in the future? This new NBER Working Paper by Diane Coffey, Michael Geruso, Dean Spears (all University of Texas at Austin), and Sangita Vyas (City University of New York), digs into the data to provide an answer. Specifically, they look at data on:

...period total fertility rates (TFRs) for all country-years since 1950 in the United Nations World Population Prospects, in completed cohort fertility for country-cohorts in the Human Fertility Database, and in fertility measures from subnational geographies in India, Japan, and the United States.

It is worth taking a moment to explain the difference between a period total fertility rate and a cohort total fertility rate. Most of the time, the statistics we see reported are period total fertility rates. Those are the number of births a woman would have across their lifetime if they experienced the age-specific fertility rate at each age in that year. Of course, each woman only experiences the age-specific fertility rate of one age in any one year, not all of them collectively. That brings us to the cohort total fertility rate, which is the average number of births each age-cohort of women (born in the same year) actually have across their lifetime. Completed cohort fertility is computed after an age cohort of women has effectively moved out of childbearing age, which is why this statistic isn't reported often - you have to wait until women are over 50 years old before you can be confident that you have an accurate measure of the cohort total fertility rate. This distinction between the two measures matters because period fertility rates can decline if women delay having children, even though the cohort fertility rate (the average number of children women have over their childbearing years) remains unchanged.

Now, coming back to the working paper by Coffey et al., they use phase diagrams to illustrate the trajectory of total fertility rates over time. A phase diagram graphs the change in a variable against the level of the variable. It essentially shows, for each level of the variable, whether the variable is increasing, or decreasing. If a total fertility rate is to stabilise at some level, then the phase diagram should show that for low levels of the rate, the change is positive, while for high levels of the rate, the change is negative. The level where the change is equal to zero is the level at which the rate would stabilise. In other words, for fertility rates to stabilise at 1.6 births per woman, we would expect them to increase if the rate is below 1.6 and decrease if the rate is above 1.6.

What do Coffey et al. find? The results for past data from the UN World Population Prospects is summarised in Figure 3 from the paper:

Notice that for most levels of the total fertility rate, the point estimate of the change (growth) in the rate is negative. The most charitable interpretation of these results is that maybe there is some weak evidence for a cross-over at a total fertility rate of between the 1.25-1.35 bin and the 1.35-1.45 bin. However, none of the estimates below the 1.45-1.55 bin are statistically significantly different from zero, and the point estimates at the lowest total fertility rates are negative.

Moving onto data from the Human Fertility Database, the results are summarised in Figure 4 from the paper:

In this case, there is no evidence to support a stabilisation of cohort total fertility rates in these data. All point estimates are negative, and all of those from the 1.45-1.55 bin and above, are statistically significantly below zero. The results for Japanese prefectures since 2010, US states since 2010, and Indian districts (based on data from the Demographic and Health Survey 2019-2021) all similarly show negative and statistically significant change in total fertility rates across almost all levels of observed total fertility rates (the one exception is the 1.05-1.15 bin for Japanese prefectures, but that result is positive but statistically insignificant).

Taken altogether, these results demonstrate that there is no evidence that declining fertility rates will reverse course, at least not on average across countries, or across subnational areas in the three countries Coffey et al. looked at. There isn't even any evidence that total fertility rates will stabilise, at least not at the levels that have been observed so far.

Low and declining total fertility rates are the norm across developed countries, and increasingly in developing countries as well. They aren't returning to replacement levels. That doesn't mean that fertility rates will decline indefinitely, but it does mean that assuming they will stabilise seems unlikely to be borne out. So, policymakers might be better off planning for continuing low fertility, rather than blaming pet ownership and waiting for a rebound.

[HT: Marginal Revolution]

Read more:

• The pandemic 'baby bust' in other countries
• Cohort effects and the downturn in US fertility since the Great Recession
• The new economics of fertility
• The economics of the falling total fertility rate in New Zealand
• The economics of fertility in high-income countries
• Can fertility return to replacement levels?

Pets and children may be complements, not substitutes

When you see people treating their fur-babies like children, you might be tempted to wonder to what extent people substitute pet ownership for raising children. You wouldn't be alone in considering the relationship between pet ownership and raising children. In 2022, Pope Francis decried the low global birthrate, noting that "dogs and cats take the place of children". But are dogs and cats really taking the place of children?

A new working paper by Kuan-Ming Chen, Ming-Jen Lin (both National Taiwan University), Hau-Hung Yang (Academia Sinica), and Shirley Yu-Hsuan Yen (National Taipei University) provides an answer, using data from Taiwan. They take advantage of dog and cat registry data from 2009 to 2020, matched to tax administration records that identify families and household composition.

To work out whether two goods (in this case, children and pets) are substitutes or complements, we need to look at the cross-price elasticity of demand, which relates the percentage change in quantity demanded of one good to a change in the price of another good. When the cross-price elasticity is a positive number, the two goods are substitutes. When the cross-price elasticity is a negative number, the two goods are complements.

Now, there are two interesting things about this paper that would be familiar to intermediate microeconomics students. First, Chen et al. are careful to distinguish between Marshallian cross-price elasticities (named after late 19th Century economist Alfred Marshall) and Hicksian (compensated) cross-price elasticities (named after 1972 Nobel Prize winner John Hicks). The difference between the two is that Marshallian cross-price elasticities combine both the income and substitution effects, while Hicksian cross-price elasticities look at only the substitution effect (see this post for more on the difference between an income effect and a substitution effect).

The second interesting thing is that Chen et al. effectively use a Slutsky decomposition to identify the cross-price elasticity of children to a change in the price of pets. This essentially involves two equations. The first equation is Slutsky symmetry:

The left-hand-side variables are the share of household spending on pets (spet) multiplied by the Marshallian cross-price elasticity of pets to a change in the 'price' of children. The first term on the right-hand side is the share of household spending on children (schild) multiplied by the Marshallian cross-price elasticity of children to a change in the 'price' of pets. The second term on the right-hand side multiplies the two shares of household spending by the difference in the income elasticity of demand for children and pets.

The second equation is much simpler. The Hicksian cross-price elasticity of children to a change in the price of pets is equal to the equivalent Marshallian cross-price elasticity plus the share of spending on pets multiplied by the income elasticity of demand for children. In other words, this equation adjusts the Marshallian elasticity to remove the income effect, leaving just the substitution effect.

So, the Hicksian cross-price elasticity of children to a change in the price of pets can be calculated using information about all of the other variables in those two equations. And gathering estimates of those other variables is what Chen et al. set about doing.

First, to estimate the cross-price elasticity of pets to a change in the 'price' of children, Chen et al. look at how increases in the 'child bonus' paid by local municipalities affect the registration of pets, since the child bonus affects the up-front cost (or 'price') of having children. Using an event study, they find that:

Both cats and dogs respond, and they respond in an ordered way. In the first post-treatment year, new dog acquisitions rise by 0.067 percentage points on a pre-treatment base of 0.332 percentage points, a 20.1% increase (𝑡 = 2.40); new cat acquisitions rise by 0.0136 percentage points on a base of 0.125 percentage points, a 10.9% increase (𝑡 = 2.36). Cats are therefore complements with children as well, but the proportional response is roughly half as large as for dogs.

...the implied Marshallian cross-price elasticity for cats is approximately −0.68, against −1.24 for dogs.

Remember that the Marshallian cross-price elasticity doesn't remove the income effect of the price change. To calculate the Hicksian cross-price elasticities, Chen et al. therefore need to estimate the other parts of the equation. So, next they estimate the income elasticities of demand for children and pets, making use of an interesting lottery:

Taiwan’s sales tax system provides a natural experiment for income shocks through its uniform invoice lottery program. As part of a nationwide effort to reduce tax evasion, nearly all consumption receipts are eligible for bi-monthly lottery drawings, with prizes reaching up to NTD10 million. Receipt holders are randomly selected, and prizes are distributed regardless of income level or background. These awards thus constitute plausibly exogenous, positive income shocks.

This lottery had already been used by other researchers in an event study (where the treatment is receiving a lottery windfall of at least NTD2 million) to estimate the income elasticity of children, finding it to be 0.15, meaning that children are a normal good. Following the same approach, but looking at the income elasticity of pets, Chen et al. find that the income elasticity of dogs is 0.11. Annoyingly, they don't report the income elasticity of cats (although they use it in their calculations later), although their other estimates suggest that it may be close to zero.

Finally, Chen et al. estimate the share of expenditure on children and pets. Using data from the Family Income and Expenditure Survey, they find the share of household spending on pets is 0.032 (about 3.2 percent). For children, they use an 'equivalence scale approach', which weights children in adult-equivalent terms at 0.3, and noting that would imply a share of spending in a three-person household of about 17 percent, they adopt a conservative share of household spending on children of 0.2.

Now, with all of the pieces together, Chen et al. move on to estimating the Hicksian cross-price elasticity, finding that it is equal to -0.198 for dogs, and -0.089 for cats. In other words, dogs and children are complements, and so are cats and children. The results are also reasonably robust to various alternative assumptions about the income elasticities and expenditure shares.

Chen et al. then go on to explore why pets and children might be complements. By looking at the timing of pet ownership and childbearing, they identify that it is much more common for families to have pets first, and children second. They infer from this that:

Pets may serve as a “starter family,” letting prospective parents rehearse caregiving and learn about their own taste for family life before a birth.

Of course, the mechanism results do not eliminate other explanations. The results are consistent with the 'starter family' channel, but it is also possible that people who are already planning to have children are more likely to register a pet beforehand. Also, these results are based on Taiwan. Perhaps pets and children are substitutes in other countries - we won't know unless further research is done. Nevertheless, these results are a good place to start, and should make us really question whether dogs and cats are taking the place of children.

[HT: Jacques Poot]

William Nordhaus, 1941-2026

I was saddened to hear of the passing of William Nordhaus last Friday. Nordhaus won the Nobel Prize in 2018 "for integrating climate change into long-run macroeconomic analysis", and it is his work on climate change and economics that he might be best known for. However, that was far from his only contribution to economics. My ECONS102 students might recognise Nordhaus as one of the first economists to formally analyse the tradeoff between strong intellectual property rights, which provide incentives for creation, but lead to under-consumption, and weak intellectual property rights, which can lead to optimal consumption, but lead to under-investment in idea creation (for more on that tradeoff, see this post).

My favourite Nordhaus contribution was actually some work he did on the change in the cost of light over time. The original research paper is available here, but the story is best told in this article from the BBC:

Back in the mid-1990s, an economist called William Nordhaus conducted a series of simple experiments with light.

First, he used a prehistoric technology: he lit a wood fire.

But Prof Nordhaus also had a piece of hi-tech equipment with him - a Minolta light meter.

He burned 20lb (9kg) of wood, kept track of how long it burned for and carefully recorded the dim, flickering firelight with his meter.

Next, he bought a Roman oil lamp, fitted it with a wick, and filled it with cold-pressed sesame oil.

He lit the lamp and watched the oil burn down, again using the light meter to measure its soft, even glow.

Bill Nordhaus's open wood fire had burned for just three hours on 9kg of wood.

But a mere eggcup of oil burned all day, and more brightly and controllably.

Why did he do this?

He wanted to understand the economic significance of the light bulb.

His research showed just how dramatic the fall in the cost of light was. Drawing on his work, the same BBC article notes that 60 hours of labour:

...had once produced the equivalent of 54 minutes of quality light [but] now produced 52 years.

That fact demonstrates just how powerful technological progress and economic growth have been over the very long run. But Nordhaus's deeper insight was that simply looking at prices and economic growth alone can lead us to miss the enormous improvements in living standards. The significance of that cannot be overstated.

I'm surprised that there doesn't appear to have been much coverage of Nordhaus's passing. Nevertheless, Yale News has a good obituary. And while searching for some others, I ran across this podcast from earlier last month, that neatly covers his life in the way a good obituary would.

Nordhaus's work extended far beyond the climate economics he is most known for. His research on the cost of light is a reminder that seemingly ordinary things can nevertheless tell important stories. He will be missed.

Is it better to publish more papers, or longer papers?

Academics get rewarded for their research output. More papers, published in higher-ranked journals, is a large part of the metric against which academics are judged. Notice that there are two aspects to that metric though - quantity (more papers) and quality (higher-ranked journals). Since it usually requires more effort to successfully publish a paper in a higher-ranked journal, is there a trade-off between quantity and quality of publications?

This 2014 article by my colleague John Gibson, published in the journal Economics Letters (sorry, I don't see an ungated version online), asks a related question about how we measure the quantity of research output - is it better to publish one long paper, or several papers that in total have the same number of pages? To answer that question, Gibson links the 2010 salaries of 223 economists from the University of California system to their publication histories (in terms of the number of publications, the quality of journals they were published in, and the number of publication pages), while controlling for seniority, experience, gender, campus, and whether the economist held a named chair or was a Nobel Prize winner.

Gibson notes that there is a wide range of journal ranking and weighting schemes, so he applies nine different ones. He finds that:

Six of the nine equations show a significant positive effect on salary from publishing more articles, conditional on the total number of quality-adjusted pages. The size of the effect is not trivial; on average, a standard deviation increase in the count of articles holding pages constant increases salary by 15 log points.

Ok, so publishing more articles is associated with a higher salary. And the difference is not small - 15 log points is equivalent to a salary increase of about 16 percent. What about more pages? On this, Gibson finds that:

...if an econometric ‘horse race’ is run between total pages and counts of articles, Vuong non-nested tests favor the model using total pages as being closer to the truth, even in the regressions where counts and pages are both statistically significant.

In other words, the total number of pages published seems to be a better predictor of academic salary than the total number of articles published, even though the number of articles also matters when holding the number of pages constant. How much does this difference matter? Gibson does some back-of-the-envelope calculations, finding that:

The predicted salary with a [Journal of the European Economic Association] article added to lifetime output is $195,980. Adding just as many pages to the literature with three Economics Letters articles increases predicted salary to $196,970—an annual gain over the single-article strategy of $990. This average economist is 16 years into their career, so with a 40-year career and five percent discount rate, the net present value of publishing three Economics Letters rather than one JEEA is $13,700.

Of course, these results are correlations rather than causal estimates, so we can't conclude that splitting a long paper into several shorter ones would necessarily increase an economist's salary. Nevertheless, the results imply that not only does quantity of research seem to matter, but how that quantity is broken up matters too. So, having lots of small articles published may lead to a higher salary for academic economists. That conclusion aside, one of the best aspects of this paper, you may have noticed, is that Gibson published it in... Economics Letters.