We sink a lot of resources into our children, be it in time or money. while they are children, the return is largely non-material. Thereafter, there can be a material return, directly if children support their parents during old age (and often the support still goes the other way), or indirectly if there is a pay-as-you-go social security system. Given this lack if private return in children, and that it is lower than the social return, there is underinvestment in children, in quantity and quality. This can be corrected with subsidized schooling and fertility incentives.
Alice Schoonbroodt and Michèle Tertilt make the argument that all this could easily be resolved by giving parents the full rights on the income of their offspring. Then they would get the full return of their investment and have socially optimal fertility and investment. Of course, this is not possible, first because of human rights, and second because children cannot be committed to give up their income. But there was a time where parents could exert such control, and then fertility was also higher. In other word, altruism is not the only reason for fertility.
This brings another interesting aspect on the paper. Under normal circumstances, the Coase Theorem would apply, but here it cannot as parents cannot contract with unborn children, and the latter cannot make promises to their parents. This inefficiency can only be resolved with public intervention, either by by giving property rights to parents, or by giving parents appropriate incentives, financed by taxing children. The fact that fertility is endogenous, contrarily to what is typically assumed, has another consquence: the standard results that allocations are efficient if the interest rate is higher than the population growth rate is not necessarily valid, because of under-fertility in addition to the usual problem of over-saving. In other word, endogenous fertility is very important, and all these social security models that just assume fertility is reduced are missing something big.