Efficiency and Endogenous Fertility

This paper explores the properties of several notions of efficiency (A−efficiency, P−efficiency and Millian efficiency) to evaluate allocations in a general overlapping generations setting with endogenous fertility and descendant altruism that includes, as a particular case, Barro and Becker’s (1988) model of fertility choice. We first focus on the notion of A−efficiency, proposed by Golosov, Jones and Tertilt (Econometrica, 2007) and show that, in many environments, the set of symmetric, interior, A−efficient allocations is empty. To overcome this problem, we then propose to evaluate the efficiency of a given allocation with a particular specification of P−efficiency –proposed also by Golosov et al.– for which the utility attributed to the unborn depends on the utility level achieved by those who get to be born in a given allocation. For a large class of specifications of the function determining the utility attributed to the unborn, every Millian efficient allocation, that is, every symmetric allocation that is not A−dominated by any other symmetric allocation, is also P-efficient. Finally, we restate the First Welfare Theorem by showing that a) every competitive equilibrium is a –statically– Millian efficient allocation; and that b) if long-run wages do not exceed the capitalized costs of rearing children, then competitive equilibria are also –dynamically– Millian efficient.