Memo_稳定人口各指标变动影响

按查瑞传《数理人口学》、凯菲兹《应用数理人口学》，就稳定人口各指标变动的推导作简要记录。

一、稳定人口基本方程

$$\begin{aligned} &b=\frac{1}{\int_0^\omega e^{-r a} l(a) d a} \qquad(1) \\ &c(a)=b e^{-r a} l(a) \qquad(2) \\ &\psi(a)=\int_\alpha^\beta e^{-r a} l(a) \phi(a) d a=1 \qquad(3) \\ &l(a)=e^{-\int_0^a \mu(a) d a} \qquad(4) \end{aligned}$$

二、$\phi(a)$ 变动影响

1. $\phi(a)$ 等比例变动对$r$的影响

$$\begin{aligned} &\phi^{\prime}(a)=e^{k a} \phi(a) \\ &1=\int_\alpha^\beta e^{-r^{\prime} a} l(a) \phi^{\prime}(a) d a \\ &\quad\ \ =\int_\alpha^\beta e^{-r^{\prime} a} l(a) \phi(a) e^{k a} d a \\ &\quad\ \ =\int_\alpha^\beta e^{-\left(r^{\prime}-k\right) a} l(a) \phi(a) d a \\ &1=\int_\alpha^\beta e^{-r a} l(a) \phi(a) d a=1 \qquad(3) \\ &-\left(r^{\prime}-k\right)=r \\ &r^{\prime}=r+k \\ &\phi^{\prime}(a) \uparrow \quad k>0 \quad r^{\prime} \uparrow \\ &\phi^{\prime}(a) \downarrow \quad k<0 \quad r^{\prime} \downarrow \end{aligned}$$

2. $\phi(a)$ 某一岁变动对$r$的影响

$$\begin{aligned} &e^x \doteq 1+x ,\quad e^{-\Delta r a} \doteq (1-\Delta r \cdot a) \\ &e^{-(r+\Delta r) a} = e^{-r a} \cdot e^{-\Delta r a} \\ &\quad\ \ = e^{-r a}(1-\Delta r a) \end{aligned}$$

$$\begin{aligned} &1=\int_\alpha^\beta e^{-(r+\Delta r) a} l(a)[m(a)+\Delta m(x)] d a \\ &\quad\ \ =\int_\alpha^\beta e^{-(r+\Delta r) a} l(a) m(a) d a+e^{-(r+\Delta r) x} l(x)\Delta m(x) \\ &\quad\ \ =\int_\alpha^\beta e^{-r a}(1-\Delta r a) l(a) m(a) d a+C \\ &\quad\ \ =\int_\alpha^\beta e^{-r a} l(a) m(a) d a-\Delta r \int_\alpha^\beta e^{-r a} l(a) m(a) a+C \end{aligned}$$

$$\begin{aligned} &1-\Delta r \cdot \mu+C=1 \\ &C=\Delta r \cdot \mu \\ &\Delta r \cdot \mu = e^{-(r+\Delta r) x} l(x) \Delta m(x) \\ &\quad\ \ = e^{-r x}(1-\Delta r x) l(x) \Delta m(x) \\ &\quad\ \ = e^{-r x} l(x) \Delta m(x) \\ &\Delta r =\frac{e^{-r x} l(x) \Delta m(x)}{\mu} \end{aligned}$$

此外有$l(a)$、$NRR$、$T$等变动影响，余略。参见笔记。