Efficiency and endogenous fertility

This paper explores the properties of the notions of A-efficiency and P-efficiency, proposed by Golosov, Jones and Tertilt (Econometrica, 2007), to evaluate allocations in a general overlapping generations setting in which fertility choices are endogenously selected from a continuum and any two agents of the same generation are identical. First, we show that the properties of A-efficient allocations vary depending on the criterion used to identify potential agents. If one identifies potential agents by their position in their siblings' birth order --as Golosov, Jones and Tertilt do--, then A-efficiency requires that a positive measure of agents use most of their endowment to maximize the utility of the dynasty head, which, in environments with finite horizon altruism, implies that some agents --the youngest in every family-- obtain an arbitrary low income to finance their own consumption and fertility plans. If potential agents are identified by the dates in which they may be born, then A-efficiency reduces to dynastic maximization, which, in environments with finite horizon altruism, drives the economy to a collapse in finite time. To deal with situations, like those arising in economies with finite horizon altruism, in which A-efficiency may be in conflict with individual rights, we propose to evaluate the efficiency of a given allocation with a particular class of specifications of P-efficiency, for which the utility attributed to the unborn depends on the utility obtained by their living siblings. Under certain concavity assumptions on value functions, we also characterize every symmetric, P-efficient allocation as a Millian efficient allocation, that is, as a symmetric allocation that is not A-dominated --with the Birth-Order criterion-- by any other symmetric allocation.