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^{eta}, quad 0<alpha, eta<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)
( ightarrowfrac{mathrm{d} y}{mathrm{d} t}+mu y=f_{0} lambda y^{alpha})
( ightarrow y(t)=left(frac{f_{0} lambda}{mu}+left(y_{0}^{1-alpha}-frac{f_{0} lambda}{mu} ight) mathrm{e}^{-(1-alpha) mu t} ight)^{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})

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