Pension and children: Pareto improvement with heterogeneous preferences

In an overlapping-generations model with endogenous birth rates, I design a reform of the pay-as-you-go pension system, which internalises positive externalities of children - their pension contributions. Individuals may differ in their preferences for children and their ability to have children at all. They can choose between the status-quo flat-rate benefits and a new system, in which they get just the benefits that are (on average) financed by their own children, reduced by an amount which is used to subsidise the flat-rate system. Whereas people with low child preferences keep the status quo, people with high child preferences choose the individualised system, having the optimal incentives to raise children and a higher utility.