Optimal fertility ditribution

In a risk-sharing dynastic model, Hosseini, Jones, and Shourideh (2009) find that fertility should be negatively related to ability. However, they only consider the case where ability is i.i.d. across generations and restrict attention to the case where children cost only time (i.e. cost proportional to productivity). These notes ask if this result may be mitigated or even overturned in various settings, if there is persistence in intergenerational ability and/or some of the costs of children are in terms of goods (i.e. not proportional to productivity). It turns out that in Barro-Becker type models (i.e. where parents care about the utility from children) if children only cost time, the negative ability-fertility relationship remains even if shocks are perfectly persistent. If children cost only goods, then the optimal relationship is positive with persistence and flat with i.i.d. shocks. So, in the simplest case, it is a mix of the relative sizes of goods and time costs together with the degree of persistence which determine what the optimal fertility distribution looks like.