Does paternity leave promote gender equality within households?

We consider a non-cooperative model of the household, in which the husband and wife decide on parental leave and the allocation of time between child rearing and the labor market. They can choose the non-cooperative outside option or cooperate by reaching an agreement of specialization in which the wife specializes in raising kids (home production) while the husband works and transfers consumption to his wife. The model identifies three distinct groups of couples: Egalitarian couples (with a sufficiently low gender wage gap), Intermediate-gap couples (with an intermediate gender wage gap) and high-gap couples (with a sufficiently high gender wage gap). Our model predicts that while egalitarian couples never specialize and always share home production, those with intermediate and high gaps do have such an agreement. An expansion in paternity leave reduces the net benefits from the agreement and moves the intermediate-gap couples to their outside option where women work more and men do more home production. As a result, the cost of raising children increases and fertility declines. Assuming a loss of utility from children in the case of divorce, lower fertility increases the probability of divorce. Using Spanish data and RDD analysis, we confirm our model's predictions. Specifically, while we don't find systematic effects of paternity leave expansion on egalitarian and high-gap couples, we find that, among intermediate-gap couples, the two-week paternity leave introduced in 2007 resulted in a reduction in fertility by up to 60%, an increase in the probability to divorce by 37%, and an increase in father's childcare and housework time as much as 2-3 hours per day.