FATHER, mother and two children: surely the perfect family size. For those concerned, it is neither too big nor too small. For the national economy, it ensures that two new workers will replace the parents in the labour force. And eventually the children will have children of their own and keep the population stable.

For that happy state to be achieved, the “total fertility rate” (a measure used by demographers for the number of children a woman is likely to have during her childbearing years) needs to be above two: around 2.1 in the rich world and more in poorer countries, because some children, particularly in the developing world, die before adulthood. For many years the United Nations’ population forecasts—the gold standard in the demography business—have assumed that, in the long run, fertility the world over would converge on the replacement level and populations would stabilise. But fertility rates everywhere have been declining for decades. Even in Africa, where large families are still the norm, the number of children per woman in 2010-15 is forecast to fall to 4.7, compared with 5.7 in 1990-95. Global average fertility is already down to about 2.5.

In a growing number of countries the fertility rate has now fallen below replacement level (see map). In China it is around 1.5 (though official figures put it slightly higher) because of the one-child policy in force since the 1970s, which has also messed up the balance between boys and girls. For Europe as a whole it is 1.6, and well below that in several southern and eastern countries. In Japan fertility has been declining for decades, to 1.4 now, and the population is already shrinking. South Korea, at 1.3, has the lowest rate of any big country. Numbers are also slipping below replacement level in less wealthy South-East Asia. Quite soon half the world’s people will live in countries where the population is no longer reproducing itself.

This worries governments, because fewer babies mean fewer workers later on, and as people are living longer, they will have to support a growing number of pensioners. Many are trying to persuade couples to be more fruitful. The UN reckons that last year two-thirds of the countries in more developed regions had policies to raise fertility, compared with one-third in 1996. Most are in Europe, but in Asia too such measures are on the rise. They range from tax incentives and child benefits to better child-care provision and making it easier for women both to have children and to work.

In some places such policies seem to have had some impact: France and the Nordic countries, which have long had them, have near-replacement fertility levels as well as lots of mothers who hold down jobs. But Germany combines generous maternity leave with one of the lowest fertility levels in Europe (1.4), and its population is dropping. Conversely, in America the fertility rate has perked up to close to replacement level, although public policy does little to encourage motherhood. It does not even run to statutory maternity leave, which is standard in rich countries.

But is a fertility rate at replacement level the right target? In a recent study Erich Striessnig and Wolfgang Lutz, of the Vienna University of Economics and Business and the International Institute for Applied Systems Analysis in Laxenburg, Austria, argue that in predicting dependency ratios (the number of children and pensioners compared with people of working age), education should also be taken into account. And that makes optimal rates much lower than previously thought.

Not everyone of working age contributes equally to supporting the dependent population. Better-educated people are more productive and healthier, retire later and live longer. Education levels in most places have been rising and are likely to continue to do so. Using projections by age, sex and level of education for 195 countries, the demographers conclude that the highest welfare would follow from long-term fertility rates of 1.5-1.8. That excludes the effects of migration: for countries with many immigrants, the figure would be lower.

Educating more people to a higher level will be expensive, both because of the direct costs and because the better-educated start work later. But they will contribute more to the economy throughout their working lives and retire later, so the investment will pay off. Moreover, fewer people will help limit future climate change.

All this suggests that worries about falling populations are better addressed by education than by baby bonuses or tax breaks. But population policies are not all about rational economics: the world pays more attention to populous countries with sizeable armies than small ones without them. And countries that feel under threat tend to look for safety in numbers. It is no accident that, almost alone among developed countries, Israel has a fertility rate well above replacement level, at 2.9.