The Population Bomb

Population control or race to oblivion?

Paul Ehrlich, 1968

World population increased from 3 billion to 6 billion between 1960 and 1999, a doubling time of 39 years, and about half as long as the time it took to go from 1.5 billion to 3 billion. Only a dozen years passed before the seventh billion came along. But contemporary demographers agree that population growth has slowed dramatically. Projections using current fertility rates point to a doubling time of 100 years or more, perhaps even forever. One widely quoted estimate has population peaking at 9 billion in 2070.— Contemporary hand-wringing about sustainable growth notwithstanding, the population bomb has been defused—what a relief!

The question of how population growth affects living standards has both a macro side and a micro side. Macro demography traces its roots to the eighteenth-century English scholar Thomas Malthus, who argued that population size increases when food output increases, so much so that productivity gains fail to boost living standards. The unhappy Malthusian outcome is characterized by a permanent subsistence-level existence for most people. This pessimistic view of economic growth has repeatedly been falsified by history, but that hasn’t prevented it from gaining traction among latter-day doomsayers. Biologist Paul Ehrlich’s 1968 blockbuster The Population Bomb famously argued for a Malthusian scenario featuring imminent mass starvation in India. Since then, India’s population has tripled, while Indian living standards have increased markedly.—

Economists have turned a micro lens on the relationship between family size and living standards. Here, attention focuses on the ability of households of different sizes to support a comfortable standard of living. We might indeed expect increases in family size to be associated with increased poverty and reduced education—more mouths to feed means less for each—and that’s what simple correlations show. A more elaborate theoretical rationalization for this powerful relation comes from the work of the late Gary Becker and his collaborators. These studies introduced the notion of a “quantity-quality tradeoff,” the idea that reductions in family size increase parental investment in children. For example, parents with fewer children might guard their children’s health more closely and invest more in their schooling.—

On the policy side, the view that smaller families are essential for increasing living standards has motivated international agencies and many governments to promote, and occasionally even to require, smaller families. China led the way with the controversial One Child Policy, implemented in 1979. Other aggressive government-sponsored family planning efforts include a forced-sterilization program in India and the public promotion of family planning in Mexico and Indonesia. By 1990, 85% of people in the developing world lived in countries where the government considered high fertility to be a major force perpetuating poverty.—

The negative correlation between average family size and development indicators like schooling is hard to argue with. Is there a causal connection between family size and children’s education? The challenge in answering this question, as always, is the paribus – ness of the ceteris. For the most part, fertility is determined by the choices parents make.— Not surprisingly, therefore, women with large families differ in many ways from those with smaller families; they tend to be less educated, for example. And the children of less – educated mothers tend to be less educated themselves. Marked differences in observable characteristics across families of different sizes raise the red flag of selection bias. Since women with different numbers of children are so observably different, we must acknowledge the possibility of important unobserved differences associated with family size as well.

As always, the ideal solution to an omitted variables problem is random assignment. In this case, the experiment might go like this, (i) Draw a sample of families with one child, (ii) In some of these households, randomly distribute an additional child, (iii) Wait 20 years and collect data on the educational attainment of firstborns who did and did not get an extra sibling. Of course, we aren’t likely to see such an experiment any time soon. Clever masters might, however, find sources of variation that reveal the causal connection between family size and schooling without the benefit of a real experiment.


Which brings us to the question of where babies come from. As most of our readers will know, human infants are delivered to households by a long-legged, long-necked bird called a stork (though it’s a myth that the infant is dropped down the chimney—chimneys have a damper that prevents delivery of a live infant). Delivery occurs 9 months after a woman, whom we will refer to as the “mother,” declares her intention to have a child. Storks are unresponsive to the wishes of men (except when these wishes are passed on by women), so we focus here on the notional experiment from the point of view of the mother and her oldest child.

The experiment we have in mind is the addition of children to households that have one already. The first-born child is our experimental subject. The ’metrics challenge is how to generate “as good as randomly assigned” variation in family size for these subjects. Unfortunately, the Association of Stork Midwives rejects random assignment as unnatural. But storks nevertheless generate circumstantially random variation in family size by sometimes delivering more than one child in the form of twins (a consequence of the fact that storks are large and infants are small, so storks sometimes scoop multiples when picking babies in the infant storage warehouse). The fact that twins induce a family size experiment was first recognized in a pioneering study by Mark Rosenzweig and Kenneth Wolpin, who used a small sample of twins to investigate the quantity-quality trade-off in India.—

To exploit the twins experiment, we turn to a large sample from Israel, analyzed in a study of the quantity-quality tradeoff by Master Joshway, with colleagues Victor Lavy and Analia Schlosser (the “ALS study” for short).— Israel makes for an interesting case study because it has a very diverse population, including many people who were born in developing countries and into large families. About half of the Israeli Jewish population is of European ancestry, while the other half has roots in Asia or Africa. Quite a few Arabs live in Israel as well, but the data for Israeli non-Jews are less complete than for Jews. An attractive feature of the Israeli Jewish sample, besides ethnic diversity and larger families than are found in most developed countries, is the availability of information on respondents’ families of origin, including the age and sex of their siblings. This unusual data structure is the foundation of the ALS empirical strategy.

We focus here on a group of first-born adults in a random sample of men and women bom to mothers with at least two children. These firstborns have at least one younger sibling, but many have two or more. Consider a family in which the second birth is a singleton. On average, such families include 3.6 children. A second twin birth, however, increases average family size by.32, that is, by about one-third of a child. Why do twin births increase family size by a Solomonic fractional child? Many Israeli parents would like three or four children; their family size is largely unaffected by the occurrence of a multiple twin birth, since they were going to have more than two children either way. On the other hand, some families are happy with only two children. The latter group is forced to increase family size from two to three when the stork delivers twins. The one-third-of – a-child twins differential in family size reflects a difference in probabilities: the likelihood of having a third child increases from about.7 with a singleton second birth to a certainty when the second birth is multiple. The.3 figure comes from the fact that the difference between a probability of 1 and probability of.7 is.3.

A simple regression of adult firstborns’ highest grade completed on family size shows that each extra sibling is associated with a reduction of about one-quarter of a year of schooling (these results come from a model with age and sex controls). On the other hand, as the ALS study shows, even though first-born adults with second-born twin siblings were raised in larger families, they are no less educated than first-born adults in families where the second-born child was a singleton. The comparison of schooling between firstborns with twin and singleton siblings constitutes the reduced form for an IV estimate that uses twin births as an instrument for family size.

IV estimates are constructed from the ratio of reduced-form to first-stage estimates, so a reduced form of zero immediately suggests the causal effect of sibship size is also zero. The fact that the twins reduced-form and associated IV estimates are close to zero weighs against the view that a larger family of origin reduces children’s schooling. In other words, the twins experiment generates no evidence of a quantity-quality tradeoff.

Multiple births have a marked effect on family size, but the twins experiment isn’t perfect. Because the Association of Stork Midwives refuses to use random assignment, there’s some imbalance in the incidence of twinning. Multiple births are more frequent among mothers who are older and for women in some racial and ethnic groups. This potentially leads to omitted variables bias in our analysis of the twins experiment, especially if some of the characteristics that boost twinning are hard to observe and control for.— Luckily, a second fertility experiment provides evidence on the quantity – quality trade-off.

In many countries, fertility is affected by sibling sex composition. For one thing, parents often hope for a son; son preference is particularly strong in parts of Asia. In Europe, the Americas, and Israel, parents seem to care little about whether children are male or female. Rather, many parents hope for a diversified sibling-sex portfolio: Families whose first two children are both boys or both girls are more likely to have a third child. Because the sex of a newborn is essentially randomly assigned (male births occur about half the time and, in the absence of sex-selective abortion, little can be done to change this), parental preferences for mixed sibling-sex composition generate sex-mix instruments.

First-born Israeli adults who have a second-born sibling of the opposite sex grew up in households with about 3.60 children. But firstborns whose second-born sibling is of the same sex were raised in families with 3.68 children. In other words, the same-sex first stage for Israeli firstborns is about.08. As with the twins first stage, this differential reflects changes in the probability of childbearing induced by an instrument. In this case, the instrumental variable is a dummy variable that equals 1 for families whose first two children are both male or both female and equals 0 for families with one boy and one girl. While the sex-mix first stage is smaller than that arising from twinning, the number of families affected by same-sex sibships is much larger than the number of families affected by twinning. About half of all families with at least two children have either two boys or two girls at births number one and number two. By contrast, only about 1% of mothers have twins. Sibling sex composition also has a leg up on twinning in being unrelated to maternal characteristics, such as age at birth and race (as shown by ALS and in an earlier study by Master Joshway and William Evans).—

As it turns out, the educational attainment of first-born Israeli adults is unaffected by their siblings’ sex composition. For example, the average highest grade completed by firstborns from families with mixed – and same-sex sibships is about equal at 12.6. Thus, the same-sex reduced form, and therefore the corresponding IV estimates, are both zero. Like the twins experiment, fertility changes generated by differences in sibling sex composition show no evidence of a quantity-quality trade-off.

The exclusion restriction required for a causal interpretation of sex-mix IV estimates asserts that sibling sex composition matters for adult outcomes only insofar as it changes family size. Might the sex-mix of the first two children affect children’s educational outcomes for other reasons? Two boys and two girls are likely to share a bedroom longer than mixed-sex siblings, for example, and same-sex siblings may make better use of hand – me-down clothing. Such household efficiencies might make families with a same-sex sibship feel a little richer, a feeling that may ultimately increase parental investment in their children’s schooling.

Can we test the exclusion restriction? Not directly, but, as is often the case, evidence can be brought to bear on the question. For some mothers, sex composition is unlikely to affect fertility. For example, in an Israeli sample, religious women who plan to have three or more children are always-takers for sex-mix instruments. On the other hand, highly educated women, most of whom plan small families, are never-takers if their fertility behavior is unchanged by sex mix. Because the fertility of always-takers and never-takers is unchanged by sibling sex composition, any relationship between sex-mix instruments and outcomes in samples with few compliers may signal violations of the underlying exclusion restriction.

We can express this idea more formally using the representation of LATE in equation (3.2). This expression defines LATE as the ratio of reduced-form to first-stage parameters, that is:


which implies in turn that the reduced form, p, is the product of the first stage and LATE:

p = фк.

From this we conclude that in samples where the first stage, ф, is zero, the reduced form should be zero as well. On the other hand, a statistically significant reduced-form estimate with no evidence of a corresponding first stage is cause for worry, because this suggests some channel other than the treatment variable (in this case, family size) links instruments with outcomes. In this spirit, ALS identified demographic groups for which the effect of twins and sex-composition instruments on family size is small and not significantly different from zero. These “no-first-stage samples” generate no evidence of significant reduced-form effects that might signal violations of the exclusion restriction.

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