Objective: To demonstrate the analytical value of a cubic parameterization of the age curve of fertility and to explore its features, especially its usefulness in separating fertility level and fertility timing. Methods: Using mathematical analysis, the cubic fertility curve is derived and examined in both continuous and discrete forms. Results: The cubic curve for replacement level fertility is found and expressed in terms of the mean age of fertility. That baseline cubic birth rate density, proportionately adjusted for the level of fertility, is shown to plausibly fit observed birth rates and imply a new approximation for their implicit stable growth rate. Because the proposed cubic model separates the effects of fertility level (quantum) and fertility timing (tempo), it leads to new period/cohort and population momentum relationships and provides a structure for relating fertility trajectories to birth sequences in changing rate models. Contribution: The cubic parameterization can simplify the representation of age curves of fertility rates while capturing their essential features. With a proportional adjustment at all ages to reflect fertility level, the cubic model can separate level and timing effects and permit numerous analytical applications. Of note, those applications include a new and superior approach to how changes in period tempo with constant quantum affect cohort fertility.

中文翻译：

---

具有立方年龄特定出生率的生育率量子和速度（作者：Robert Schoen）


目的： 展示生育年龄曲线三次参数化的分析价值，并探讨其特征，特别是其在分离生育水平和生育时间方面的有用性。方法：使用数学分析，以连续和离散形式推导出和检查立方生育率曲线。结果： 找到更替水平生育率的三次曲线，并以平均生育年龄表示。根据生育率水平成比例调整的基线立方出生率密度被证明与观察到的出生率合理匹配，并暗示了其隐含稳定增长率的新近似值。由于所提出的三次模型将生育水平 （quantum） 和生育时间 （tempo） 的影响分开，因此它导致了新的时期/队列和人口动量关系，并提供了一种在变化率模型中将生育轨迹与出生序列相关联的结构。贡献： 三次参数化可以简化生育率年龄曲线的表示，同时捕捉其基本特征。通过对所有年龄进行比例调整以反映生育水平，三次模型可以分离水平和时间效应，并允许进行多种分析应用。值得注意的是，这些应用包括一种新的、卓越的方法，用于研究具有恒定量子的周期节奏变化如何影响队列生育率。