According to Phelps' Golden Rule, a rise in fertility decreases the optimal capital intensity, because a higher fertility increases the investment required to sustain a given capital intensity (the capital dilution effect). Using a matrix population model embedded in a two-period overlapping generation setting, we examine the robustness of that relationship to the partitioning of the population into two subpopulations having distinct fertility behaviors and entering the production process as distinct inputs. We show that, unlike what prevails under a homogeneous population, a rise in fertility (caused by a change in type-specific fertility) does not necessarily reduce the Golden Rule capital intensity. The intuition is that changes in type-specific fertility modify the composition of the labor force, which affects the marginal productivity of capital and the capital dilution effect. When the composition effect induced by the fertility change outweighs the standard capital dilution effect prevailing under a fixed partition of the population, a rise in fertility increases the optimal capital intensity. These results are robust to a finer description of heterogeneity, that is, a partitioning of the population into a larger number of subpopulations.

中文翻译：

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生育率、异质性和黄金法则

根据菲尔普斯的黄金法则，生育率的上升会降低最优资本密集度，因为较高的生育率会增加维持给定资本密集度所需的投资（资本稀释效应）。使用嵌入两期重叠一代环境中的矩阵人口模型，我们检查了将人口划分为具有不同生育行为并作为不同投入进入生产过程的两个亚群体之间的关系的稳​​健性。我们表明，与同质人口中普遍存在的情况不同，生育率的上升（由特定类型生育率的变化引起）并不一定会降低黄金法则资本密集度。直觉是，特定类型生育率的变化改变了劳动力的构成，从而影响资本的边际生产率和资本稀释效应。当生育率变化引起的构成效应超过人口固定分配下普遍存在的标准资本稀释效应时，生育率的上升会增加最优资本密集度。这些结果对于异质性的更精细描述是稳健的，即将群体划分为更多数量的亚群体。