Evolutionary selection among utility functions and preferences for fertility

Utility functions that emphasize preferences for grandkids are more consistent with the theory of biological evolution.

Summary

How should economists model fertility preferences? Evolution gives a different answer to this question than economists “intuition”.

Evolution selects for people who prefer to have more children (q). Depending on the prevailing technology of production, it may also select people who also prefer to invest in the quality (education) of children (z). But evolutionary selection implies even then these investments are just an indirect input into quantity of children (q(z)). The same logic applies to other consumption (c).

This logic suggests that, in the long run, the most stable utility function posits that utility (u) is directly only a function of quantity (q), and perhaps indirectly a function of quality (z) of children and other private consumption (c), both of which can be thought of as inputs into the quantity of children: u(q(c,z)). Such preferences substantially differ from the usual economic models, which posit utility as a direct function of consumption and perhaps the quality and quality of children: u(c,q,z).

Background

Social science theories are shaped by their environment. For example, Malthusian (or classical) economics was birthed in a context in which a larger population correlated with lower per capita (caloric) consumption, and Marxism was inspired by the poor conditions experienced by workers migrating to urban areas to work in early Industrial Revolution factories. Likewise, new-classical economics emerged to explain markets and trade, largely in a post-Industrial Revolution context.

The theory that emerged had as its lynchpin a utility function (u) over consumption of private¹ goods (a vector c), including food:

U(c).

Combined with a budget constraint such as pc = I, where I is income, it produced demand functions that had great predictive power.

The problem with this simple model is that it ignored the size of the population, an important outcome. Part of the problem was positive: how did people choose whether to sacrifice consumption to have more kids, clearly a common and important behavior? Part was normative: should a utilitarian prefer a larger population with less per capita consumption (and utility) or a smaller population with greater consumption? I will focus on the positive aspect in this essay.

Gary Becker, most notably, took up the economic problem of fertility. His theory posited a modification of the utility function to account for the value and cost of children. The simplest of these models began with people obtaining utility from both private consumption and the quantity of children q,

U(c, q),

and with a budget constraint that included the cost of children, p c + q = I, where prices are normalized by the price of having children (so the price of children is 1). Under this theory, greater income would lead to greater fertility, consistent with observation not just in Malthusian times, but also during the first century of the Industrial Revolution.

The problem with this simple formulation is that it fails to capture the demographic transition, which suggested a negative relationship between income and fertility: as incomes rose, we observed that countries’ fertility declined. Europe began the transition in the very late 1800s and the US in the early 1900s. Becker knew this, as he was writing in the 1960s and 1970s. So he posited a more sophisticated model in which people also had preferences over the quality of children,

U(c, q, z)

and a budget constraint that included quality, p C + z q = I. Quality in this simple model was measured by the amount spent on each child. In this model increased income produces greater demand for quantity and quality of children, but preferences for quality reduce the relationship between income and fertility. Indeed, depending on the shape of preferences, i.e., how fast marginal utility diminishes with quantity versus with quality of children, this model could lead to lower fertility with income.

Subsequent scholars, such as Oded Galor, probed why people had preferences for quality. The worry is that adding quality to preferences is a just-so adjustment that explains history but does not predict the future. For example, this model has trouble explaining why, although countries go through the demographic transition as incomes rise, they begin and end the transition at different levels of per-capita income. Galor’s answer (Unified Growth Theory) was that individuals vary in their preferences and differential fertility selects not just genes but also preferences. Specifically, if one examines a labor market that rewards human capital, a common trait as economies develop, then individuals who have preferences not just for the quantity of children, but also the quality of children, will actually have higher income and thus overall fertility. As a result, these will become a bigger fraction of the population. The result will be a both a lower fertility rate and a high prevalence of preference for quality of children.

Evolutionarily-robust preferences

The important insight of Galor is not merely explaining why Becker’s model is consistent with evolution, but that evolution operates on observed preferences. Evolutionary theory posits that individuals vary in genes, replication cause small changes in those genes, and the environment (competition for resources) selects (at the population level) for the most prolific subset of those mutations. By the same token, it could also posit that individuals vary in preferences, replication causes small changes in preferences among our children, and reproduction selects for the most prolific of those preferences.

However, if we take the evolutionary theory of preferences seriously, we would draw two conclusions at odds with Becker’s theory. First, individual preferences for the quality of children is a by-product of technology. In a world where the returns to human capital are growing over time, selection may prefer preferences for the quality of children. But in a world where, for example, artificial intelligence (AI) is a substitute for human capital or where, because of the welfare state, income is not a constraint on reproduction, one could imagine returning to a world in which preferences for quality of children disappear.

One might ask: can’t the same can be said for preferences for quantity of children? After all, AI, combined with robotics, can also substitute for the quantity of children as well. And welfare might make it so that one does not need children for old-age support.

Our second conclusion answers this question in the negative: evolution will select against preferences that do not include the quantity of children. If some individuals choose not to reproduce, their preferences will also not reproduce. Indeed, in the long-run, evolution suggests that the only technology-robust model of preferences has utility derived directly only from quantity of children. Quality of children, and even consumption only indirectly affect utility because they are merely inputs into the quantity of children: u(q) where q is given by the survival function q(z,c), or

u(q(z,c))

An important implication of this model is that the marginal utility of consumption importantly depends on the marginal utility of fertility. Whereas the first-order conditions for the standard model (u(c,q,z) subject to pC + zq = I) implies setting the marginal rate of substitution between consumption and kids in the utility function equal to the the marginal rate of transformation (p/z)

(du/dc) / (du/dq) = p/z,

the technology-robust preference first-order conditions imply that the marginal rate of substitution of the survival function is equal to the marginal rate of transformation:

(du/dq) (dq/dc) / (du/dq) = dq/dc = p/z.

That is, what matters for consumption choices is how it produces more fertility, not how it directly produces more utility. All talk of diminishing marginal utility of consumption is replaced with diminishing marginal fertility from consumption. This is an important restriction on utility functions, akin to using Lancaster utility function to constrain predictions in home production models.

Why does this matter?

First, it provides structure that allows a greater connection of evolutionary biology and economics. Right now, economic theorizing is grounded in observation (revealed preference). But this is limited by the fact that our observations are time-contingent. We have a lot more data on the last 50 years than the 200 years before. And vastly more on the 250 years since the Industrial Revolution than the 1000, 5000 or even 10,000 years since the Neolithic Revolution, and vastly more on the last 10,000 years than on the ~300,000 years since sapiens appeared, and so on.

Yet, the importance of evolution rises with time-span, i.e., our hunter-gathering roots before the Neolithic Revolution are far more important to our evolution (including preferences) than the last 250 or 50 years. Moreover, there are remarkable similarities between humans and other species, especially our ancestors (early homo), species with a common ancestor (apes), and perhaps mammals generally. That is, the data modern economics focuses on is unable to connect economics and evolution, two of the most successful theories about human function of the last two centuries.

Second, changing our model of human preferences may help us address other puzzles in human preferences. Let me give an example. From the population-level perspective, the modal preference depends on the relative frequency of preferences in a population. Thus, this selection-on-preferences approach to utility functions may provide a microfoundation for relative preferences. The approach may explain puzzles such as why the top 1% of income in both the years 1900 and 2000 manifest declining returns to consumption, even though the top 1% of 1900 have less income than the bottom 5% in 2000. If individuals have utility over consumption and utility is stable over time, then the 1% in 1900 should not manifest diminishing marginal utility of income (or the bottom 5% in 2000 should manifest diminished marginal utility of income. By contrast, if evolution selects — at the population level — for those with the highest marginal fertility from consumption, it will over time shift upwards the minimum consumption required to achieve a given increment of fertility. In short, selection may be able to explain why marginal utility diminishes based on the percentile of a person’s income rather than the level of that income.

Limitations

There are (at least) three important limitations to the proposition that, in the long run, only preferences for quality are direct and preferences for consumption or quality are mediated by preferences for fertility.

The first is that humans live long after they stop reproducing. In that context, consumption may have value, even though it does not produce more children.

I conjecture this is a second-order effect for two reasons. One is that, so long as extended families remain intact, adults who can no longer have children can use consumption to care for grandchildren. Indeed, this limitation actually suggests that evolution selects not so much for preferences for children as preferences for grandchildren. The other reason is that, only the elderly seem to prefer consumption without producing concomitant fertility, i.e., appear as if their preferences were u(c) only.² The importance of such direct utility from consumption to population-level preferences is proportional to the share of the population that is above reproductive age. So long as fertility remains high enough, this will be a small part of the population. Because there is selection based on fertility, one should expect fertility not to fall too much over the long run.

A second limitation is that my model of selection-of-preference implicitly assumes preferences are handed down from parent to child, with some minor variation or mutation. It is possible that, not just technology, but that culture — including actors like the state and perhaps even the entertainment industry — may attenuate the parent-child transmission of values and push children towards preference for lower fertility. Unless this cultural hijacking is thorough or permanent, it merely produces a short- to medium-term suppression of fertility. In the long run, pockets of people with a preference for high fertility will become a larger fraction of the population. Those preferences will once again become the modal preference.

Third, my proposed preference structure cannot obviously explain why fertility ever falls when income rises. If utility (u) is monotonically increasing in fertility (q), then increased income should increase fertility. Yet, throughout the world, we observe that median incomes are rising but fertility is falling. This leaves us two options. One, reject my preference structure. Two, refine it. If we want to connect evolution to economics, the latter approach is judicious. For example, perhaps utility is concave in fertility, but not always increasing. So utility is u-shaped in number of kids, so there is an “optimal” fertility. In other words, humans have ideal fertility preferences. (In a later Substack post or working paper, I will give a more formal example supported by data.)³

Conclusion

I am not entirely sure my approach to utility from quality and consumption, that preferences for these items is derivative of preferences for fertility, is entirely correct. But it is an avenue worth exploring. If we find that it is inconsistent with data, we might have to adapt it to match data over the long run. For example, it may be that utility is u(q(z,c),c).

Some might object that the usual formulation of utility renders this inquiry irrelevant. Specifically, u(q,z,c) captures all possible dynamics. I disagree. First, this traditional formulation has restrictions (i.e., utility is concave increasing in its elements) that make it so that all relationships between fertility q and other items z or c are relegated to cross-elasticities in utility. But those elasticities are under-theorized. My proposal can be interpreted as suggesting we be explicit about the theoretical structure for cross-elasticities. Second, the traditional model’s abstraction does not help us make connections between evolution and economics. It focuses entirely on the available data. And we tend to focus on recent data, not pre-historic or non-human data, which is what we must do to connect evolution and economics.

1. Later economics would allow for consumption of social goods, such as other peoples’ welfare. Except for the narrow case where the other people are children, the private versus social distinction is not important for this essay.

2. The elderly seem to possess a demand for consumption without concomitant fertility in the same way a human has a tail bone that was necessary only to evolutionary ancestors but remains because it had no private fitness cost.

3. One might be worried that this implies that at some level utility must also fall in consumption. But, because of free disposal, this is not the case. Humans can choose not to consume. This isn’t an option with children in any society where infanticide is socially punished.