Assessing altruism

Is intergenerational altruism 0.09?

In the standard Barro-Becker model of fertility, utility u in period t is:

\(u_{t} = c_{t}^{\sigma} + \alpha n_{t}^{1-\epsilon}u_{t+1}\)

with c as consumption, alpha as intergenerational altruism, n as the number of children and sigma and epsilon reflecting diminishing marginal returns to material consumption and children respectively. Alexandrie and Eden 2023 find that alpha would need to be 0.09 for a steady state solution i.e. individuals must value the welfare of their first child at merely 9% of their own, given other commonly accepted parameter values.¹ This value seems paradoxical considering that the average child's consumption level is far above 9% of their parent's, while under log utility functions under altruism of 0.09 the child’s consumption would be 9% of the parent’s. Indeed, an infant's market consumption could feasibly surpass £100,000 under some definitions given that their demand for attention equates to about four full-time equivalent (FTE) units per year. So, what could possibly account for this discrepancy?

One possible explanation could lie in the scale economies in household production, allowing parents to provide certain services much below market prices. This theory holds water in scenarios like caring for infants, where a parent promising to feed a baby whenever they wake has a much lower utility cost than paying someone else to do so - except for the very wealthiest individuals. However, this doesn't fully elucidate many parental behaviours, such as the allocation of extensive resources towards toys and children-specific activities, where such household-generated economies of scale are non-existent - and it regardless appears that the marginal toy contributes significantly less than 11.1 times the utility than the same consumption would confer to a parent.

Another theory posits that children are utility monsters, an idea that, while unprovable due to its unobservable nature, could hold some merit. Children may exhibit a higher value of nu than the average adult, appearing more content with and attached to a single toy than adults are with comparable levels of wealth. This assertion seems paradoxical, as children have abundant leisure time, and if consumption returns per unit of leisure time diminish, children should have a lower value of nu. The quantification of this distinction would be intriguing, possibly using children’s risk preferences with differing amounts of sweets, although this could be complicated by the potential misunderstanding of probabilities on the children’s part due to internal computational constraints.

Yet another explanation suggests that parents view their children as a prestige good. Accordingly, children partaking in various activities becomes a costly signal, trapping us in an inefficient equilibrium. In this case, a government might - in a way similar to the optimal response if education is primarily signalling - reduce the current subsidies currently allocated to children’s consumption, for example via tax exemptions on many goods primarily intended for children, or could even tax their consumption once all existing net subsidies are removed.

A more plausible answer, perhaps, is that most individuals are not consequentialists—they don’t value the potential utility of a person who could exist but does not as equal to the utility of an existing person. This belief seems quite plausible given typical responses to philosophical questions. Such responses are generally only partially total utilitarian in nature; otherwise, many more beliefs beyond those currently expressed would follow. This would imply that consistent consequentialists would have more children than deontologists, ceteris paribus - which would be fascinating to investigate. 

1. For two situation, the implied values are even more infeasible at 7.4%. The disparity is also worse even than this because fertility rates in much of the developed world are below steady state.