Why an economy's financial development matter for its fertility? For one, if there is little in terms of savings technology, then households need to find other ways in which they can save for old age, and children have traditionally been a good way to do this. But as long as property rights are reasonably well established, this should be that important, as there are many ways beyond financial assets to accumulate wealth, such as land, real estate, jewelry and cattle. Where a financial system can really bring change is when it gives access to credit for households.
Valerio Filoso and Erasmo Papagni study this with a life-cycle model where there is altruism from parents to offspring and vice-versa. They show that all depends on whether children are an inferior or a normal good, like in poor respectively rich countries. In addition, a relaxation of borrowing constraints allows greater investment in children. There is therefore ambiguity, which is compounded by price and second order effects. To sort it all out, Filoso and Papagni use cross-country data to estimate the sum of all these effects and find indeed that financial development decreases notably fertility in poor economies and increases it in rich ones. This may be an explanation for a fact that puzzled me for some time, why the US has a higher fertility than other rich countries.