Growth Accounting 笔记
basics
\[Y=F(K,AL)\] \[Y(t)=F(K(t),A(t)L(t))\] \[\dot{Y(t)}=\frac{ \partial Y(t) }{ \partial K(t) } \dot{K(t)}+\frac{ \partial Y(t) }{ \partial L(t) } \dot{L(t)}+\frac{ \partial Y(t) }{ \partial A(t) } \dot{A(t)}\]Divide both sides by Y (t):
\[\frac{\dot{Y(t)}}{Y(t)}=\frac{K(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial K(t) } \frac{\dot{K(t)}}{{K}(t)}+\frac{L(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial L(t) } \frac{\dot{L(t)}}{{L}(t)}+\frac{A(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial A(t) } \frac{\dot{A(t)}}{{A}(t)}\]elasticity of output
\[\alpha_{K(t)}=\frac{K(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial K(t) }\] \[\alpha_{L(t)}=\frac{L(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial L(t) }\]get
\[\frac{\dot{Y(t)}}{Y(t)}=\alpha_{K(t)}\frac{\dot{K(t)}}{{K}(t)}+\alpha_{L(t)}\frac{\dot{L(t)}}{{L}(t)}+\frac{A(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial A(t) } \frac{\dot{A(t)}}{{A}(t)}\] \[R(t)=\frac{A(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial A(t) } \frac{\dot{A(t)}}{{A}(t)}\]growth in output $\frac{\dot{Y(t)}}{Y(t)}$ growth in capital $\frac{\dot{K(t)}}{{K}(t)}$ growth in labor $\frac{\dot{L (t)}}{{L}(t)}$ elasticity of output with respect to capital $\alpha_{K(t)}$ elasticity of output with respect to labor $\alpha_{L(t)}$
measurement: capital
永续存盘法
\[K(t+1)=K(t)+I(t)-\delta K(t)\] \[start \ with \ some \ K (0)\]关于资本存量的估算 张军 中美工作时间的对比 生育率和卷
measurement: output elasticities
\[\alpha_{K(t)}=\frac{K(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial K(t) }\] \[\alpha_{L(t)}=\frac{L(t)}{Y(t)}\frac{{ \partial Y(t) }}{ \partial L(t) }\]if labor and capital earn their marginal product:
\[r(t)=\frac{ \partial Y(t) }{ \partial K(t) }\] \[w(t)=\frac{ \partial Y(t) }{ \partial L(t) }\]in this case output elasticities become factor shares:
\[\alpha_{K(t)}=\frac{K(t)}{Y(t)}r(t)=S_{K}(t)\] \[\alpha_{L(t)}=\frac{L(t)}{Y(t)}w(t)=S_{L}(t)\]dual growth accounting: Hsien (2002)
start with the accounting identity:
\[Y=rK+wL\]对 t 求导，除以 y
\[\dot{Y}=\dot{r}K+\dot{K}r+\dot{w}L+\dot{L}w\] \[\frac{\dot{Y}}{Y}=\frac{K(t)}{Y(t)}r(t)\left( \frac{\dot{r}}{r}+\frac{\dot{K}}{K} \right)+\frac{L(t)}{Y(t)}w(t)\left( \frac{\dot{w}}{w}+\frac{\dot{L}}{L} \right)=S_{K}(t)\left( \frac{\dot{r}}{r}+\frac{\dot{K}}{K} \right)+S_{L}(t)\left( \frac{\dot{w}}{w}+\frac{\dot{L}}{L} \right)\] \[\frac{\dot{Y}}{Y}-S_{K}(t)\frac{\dot{K}}{K} -S_{L}(t)\frac{\dot{L}}{L}=S_{L}(t)\frac{\dot{w}}{w}+S_{K}(t)\frac{\dot{r}}{r}\]left: primal measure of R (t) right: dual measure of R (t)
references
罗默 ：
家庭效用函数为：
\[U= \int_0^{\infty} \frac{e^{-\rho t}(L(t))}{H}u(C(t))\,dt=\int_0^{\infty} e^{(n-\rho )t}u(C(t))\,dt\]given that H=1 瞬时效用函数为：
\[u(C(t))=\frac{C(t)^{1-\theta}}{1-\theta}, \theta>0,\rho-n-(1-\theta)g>0\]家庭预算约束：
\[\int_0^{\infty} \frac{e^{-\rho t}(L(t))}{H}u(C(t))\,dt\]