经济增长模型

Douglas生产函数

用\(Q(t),K(t) ,L(t)\)分别表示某一地区或部门在时刻t的产值、资金和劳动
力，它们的关系可以一般地记作
\(Q(t)=F(K(t), L(t))\)
\(z=Q / L, \quad y=K / L\)
\(z=c g(y), \quad g(y)=y^{a}, \quad 0<\alpha<1\)
\(Q=c K^{\alpha} L^{1-\alpha}, \quad 0<\alpha<1\)（Cobb-Douglas生产函数）

\(\frac{\partial Q}{\partial K}, \frac{\partial Q}{\partial L}>0, \quad \frac{\partial^{2} Q}{\partial K^{2}}, \frac{\partial^{2} Q}{\partial L^{2}}<0\)
\(\frac{K Q_{K}}{Q}=\alpha, \quad \frac{L Q_{L}}{Q}=1-\alpha, \quad K Q_{K}+L Q_{L}=Q\)
\(\alpha\)是资金在产值中占有的份额，\(1-\alpha\)是劳动力在产值中占有的份额． 于是\(\alpha\)的大小直接反映了资金、劳动力二者对于创造产值的轻重
关系
\(Q=c K^{\alpha} L^{\beta}, \quad 0<\alpha, \beta<1\)

投资增长率与产值成正比，比例系数\(\lambda\)>0, 即用一定比例扩大再生产；

劳动力的相对增长率为常数\(\mu\), ,\(\mu\) 可以是负数，表示劳动力减少．
\(\frac{\mathrm{d} L}{\mathrm{d} t}=\mu L\)

\(\frac{\mathrm{d} K}{\mathrm{d} t}=\lambda f_{0} L y^{\alpha}\)
\(\frac{\mathrm{d} K}{\mathrm{d} t}=L \frac{\mathrm{d} y}{\mathrm{d} t}+\mu L y\)
\(\rightarrow\frac{\mathrm{d} y}{\mathrm{d} t}+\mu y=f_{0} \lambda y^{\alpha}\)
\(\rightarrow y(t)=\left(\frac{f_{0} \lambda}{\mu}+\left(y_{0}^{1-\alpha}-\frac{f_{0} \lambda}{\mu}\right) \mathrm{e}^{-(1-\alpha) \mu t}\right)^{1 / 1-\alpha}\)
\(y_{0}=K_{0} / L_{0}, Q_{0}=f_{0} K_{0}^{\alpha} L_{0}^{1-\alpha}, \dot{K}_{0}=\lambda Q_{0}\)

\[\rightarrow y(t)=\left\{\frac{f_0 \lambda}{\mu}\left[1-\left(1-\mu \frac{K_{0}}{\dot{K}_{0}}\right) e^{-(1-\alpha) \mu t}\right]\right\}^{\frac{1}{1-\alpha}} \]

\(\frac{d y}{d t}+\mu y=c \lambda y^{\alpha}(0<\alpha<1)\)
解析解：

\[y(t)=\left\{\frac{c \lambda}{\mu}\left[1-\left(1-\mu \frac{K_{0}}{\dot{K}_{0}}\right) e^{-(1-\alpha) \mu t}\right]\right\}^{\frac{1}{1-\alpha}} \]

Bernoulli方程

\(\frac{\mathrm{d} y}{\mathrm{d} x}+p(x) y=q(x) y^{n}\)
两边除以\(y^n\)
\(z=y^{1-n}\)
\(\frac{\mathrm{d} z}{\mathrm{d} x}+(1-n) p(x) z=(1-n) q(x)\)

\(\frac{\mathrm{d} y}{\mathrm{d} t}+\mu y=f_{0} \lambda y^{\alpha}\)
\(\rightarrow\frac{\mathrm{d} y}{\mathrm{d} t}*y^{-\alpha}+\mu y^{1-\alpha}=f_{0} \lambda\)
\(y^{1-\alpha}=z\)
\(\frac{dz}{dt}+\mu*z=f_{0} \lambda\)