Efficiency and endogenous fertility

Abstract

This paper explores the properties of the notions of $\mathcal{A}$‐efficiency and $\mathcal{P}$‐efficiency, which were proposed by Golosov et al.. (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 $\mathcal{A}$‐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 $\mathcal{A}$‐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 $\mathcal{A}$‐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 $\mathcal{A}$‐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 $\mathcal{P}$‐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, $\mathcal{P}$‐efficient allocation as a Millian efficient allocation, that is, as a symmetric allocation that is not $\mathcal{A}$‐dominated—with the birth‐order criterion—by any other symmetric allocation.