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.

中文翻译：

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效率和内生生育能力

本文探讨了几种效率概念（A-效率、P-效率和 Millian 效率）的特性，以评估具有内生生育力和后代利他主义的一般重叠世代环境中的分配，其中包括作为特殊情况的 Barro 和 Becker (1988) ) 生育选择模型。我们首先关注 Golosov、Jones 和 Tertilt (Econometrica, 2007) 提出的 A-效率的概念，并表明，在许多环境中，对称的、内部的、A-效率分配的集合是空的。为了克服这个问题，我们然后建议使用特定的 P 效率规范来评估给定分配的效率——Golosov 等人也提出了——为此，归因于未出生者的效用取决于那些谁实现的效用水平出生在给定的分配中。对于确定归因于未出生的效用的函数的一大类规范，每个 Millian 有效分配，即每个不由任何其他对称分配 A 支配的对称分配，也是 P 有效的。最后，我们重申第一福利定理 a) 每个竞争均衡都是一个静态的米利安有效分配；b) 如果长期工资不超过养育孩子的资本化成本，那么竞争均衡也是“动态的”米利安有效的。我们通过证明 a) 每个竞争均衡都是一个静态的 Millian 有效分配来重申第一福利定理；b) 如果长期工资不超过养育孩子的资本化成本，那么竞争均衡也是“动态的”米利安有效的。我们通过证明 a) 每个竞争均衡都是一个静态的 Millian 有效分配来重申第一福利定理；b) 如果长期工资不超过养育孩子的资本化成本，那么竞争均衡也是“动态的”米利安有效的。