It is conventional wisdom in policy circles that if you want a policy intervention to benefit children, transfers have to be paid out explicitly to the mother. The understanding is that mothers care more about their children than men, and thus are more likely to use the funds for them, directly or indirectly. There is really not reason to this backfire, but as two recent papers show, it can, in fact.
Matthias Doepke and Michèle Tertilt build a series of non-cooperative bargaining models of the household and show that things can go wrong with targeted transfers or women empowerment in general. Indeed, for transfers to have an impact on the intra-household allocation of public goods, there needs to be some kind on friction. The specifics of this friction have a large impact. For example, if women are hard-wired to prefer spending on children, then transfers targeted to them may lead to over-spending on children and under-spending on other public goods that also benefit children (say, shelter), reducing child welfare. Or: if the difference between men and women is in the market wage, women will naturally tends to more time intensive activities in the household, such as child rearing. Empowering women leads them to spend less time at home, hurting the children. If empowerment implies that women have access to more private goods (such as bars or entertainment), they will focus less on public goods that also benefit children. While these examples seem a bit convoluted, they highlight that things are not so simple.
Olivier Bargain and Olivier Donni show in another series of models with altruistic parents that targeted transfers may not work as well as targeted price subsidies. They demonstrate that price subsidies have an income effect and a substitution effect, something we teach undergraduates. But they reinterpret the substitution effect as a "targeting effect." Naturally, transfers only lead to an income effect. Thus subsidies are better at improving children welfare, but they are more expensive as they apply to everyone. So it all depends on elasticities, and depending on the situation, transfers or price subsidies could be preferred.