Learning to Perfect Manipulation: Implications for Fertility, Savings, and Old-Age Social Security

In this paper we consider an overlapping gnerations model with endogenous fertility and two-sided altruism and show the limitations of applying commonly used open loop Nash equilibrium in characterizing equilibrium transfers from parents to children in the form of bequest, and transfers from children to parents as voluntary old-age support. Since in our model children are concerned with parents' old-age consumption, agents have incentives to save less for old age and to have more children so as to strategically induce their children to transfer more old-age support. We formulate such strategic behavior within a sequential multi-stage game and introduce a notion of learning equilibrium to characterize equilibrium manipulative behavior and then study the consequences of such strategic manipulations on private intergenerational transfers, fertility and savings decisions, and on Pareto optimality of equilibrium allocation. We show that the learning equilibrium notion of the paper simplifies computation of subgame perfect equilibrium, subgame perfect equilibrium is the long-run outcome of dynamic learning equilibrium paths (this aids in selecting, sometimes, a unique equilibrium among multiple subgame perfect equilibria), and an open-loop Nash equilibrium involves "incredible" threats from children. We provide an alternative explanation for the existence of publicly provided social security program and examine its role to correct distortions created by strategic manipulation.