Production function:
\[
Y=F(K,L)
\]
$\textbf{Y}$: output, $\textbf{K}$: capital stock, $\textbf{L}$: labour force
\[
k=K/L
\]
$\textbf{k}$: capital/labour ratio
\[
y=Y/L
\]
$\textbf{y}$: output/labour ratio
\[
\frac{Y}{L}=\frac{F(K,L)}{L}=F\left(\frac{K}{L},1\right)=F(k,1)=f(k)
\]
\[
y=f(k)
\]
\[
f(0)=0,\;f'(k)>0,\;f''(k)<0,\;k>0
\]
Assumptions:
\[
\dot{L}=nL,\;L(0)=L_{0}
\]
$\textbf{n}$: constant growth rate of labour force
\[
S=sY
\]
$\textbf{S}$: savings as a constant fraction of output
\[
S=I
\]
$\textbf{I}$: investment, which is the change in the capital stock plus replacement investment:
\[
I=\dot{K}+\delta K
\]
\[
K(0)=K_{0}
\]
Show Solution
Differentiating the variable $k$ with respect to time
\[
\frac{dk}{dt}=\dot{k}=\dot{\left(\frac{K}{L}\right)}=\frac{L\dot{K}-K\dot{L}}{L^{2}}=\frac{\dot{K}}{L}-\frac{K}{L}\cdot \frac{\dot{L}}{L}=\frac{K}{L}\cdot \frac{\dot{K}}{K}-\frac{K}{L}\cdot \frac{\dot{L}}{L}=k\;\left(\frac{\dot{K}}{K}-\frac{\dot{L}}{L}\right)
\]
\[
\dot{K}=I-\delta K=S-\delta K=sY-\delta K
\]
\[
\frac{\dot{K}}{K}=\frac{sY-\delta K}{K}=\frac{sY}{L}\left(\frac{L}{K}\right)-\delta =\frac{sf(k)}{k}-\delta
\]
\[
\frac{\dot{L}}{L}=\frac{nL}{L}=n
\]
\[
\dot{k}=sf(k)-\delta k-nk=sf(k)-\left(n+\delta \right)k
\]
Initial conditions:
\[
k(0)=\frac{K_{0}}{L_{0}}=k_{0}
\]
We use a Cobb-Douglas production function:
\[
Y=aK^{\alpha }L^{1-\alpha },\;0<\alpha <1
\]
\[
\frac{Y}{L}=a\left(\frac{K}{L}\right)^{\alpha }
\]
\[
y=f(k)=ak^{\alpha }
\]
\[
\dot{k}=sak^{\alpha }-\left(n+\delta \right)k
\]
\[
\dot{k}+\left(n+\delta \right)k=sak^{\alpha }
\]
This is a Bernoulli equation, which takes the general form
\[
\frac{dk}{dt}+kP(t)=k^{\alpha }Q(t)
\]
\[
k^{-\alpha }\dot{k}+k^{1-\alpha }P(t)=Q(t)
\]
\[
k^{-\alpha }\dot{k}+\left(n+\delta \right)k^{1-\alpha }=sa
\]
Defining the following transformation
\[
v=k^{1-\alpha }
\]
we obtain
\[
\dot{v}=\left(1-\alpha \right)k^{-\alpha }\dot{k}
\]
or
\[
\dot{k}=\frac{k^{\alpha }}{1-\alpha }\dot{v}
\]
\[
k^{-\alpha }\frac{k^{\alpha }}{1-\alpha }\dot{v}+\left(n+\delta \right)v=sa
\]
\[
\frac{\dot{v}}{1-\alpha }+\left(n+\delta \right)v=sa
\]
\[
\dot{v}+\left({1-\alpha }\right)\left(n+\delta \right)v=\left({1-\alpha }\right)sa
\]
which is a linear differential equation in $v$.\;Solution:
\[
v(t)=\frac{sa}{n+\delta }+\left(v_{0}-\frac{sa}{n+\delta }\right)e^{-\left({1-\alpha }\right)\left(n+\delta \right)t}
\]
which satisfies the initial condition
\[
v_{0}=k_{0}^{1-\alpha }
\]
\[
k^{1-\alpha }=\frac{sa}{n+\delta }+\left(k_{0}^{1-\alpha }-\frac{sa}{n+\delta }\right)e^{-\left({1-\alpha }\right)\left(n+\delta \right)t}
\]
Hence
\[
k(t)=\left[\frac{sa}{n+\delta }+e^{-\left({1-\alpha }\right)\left(n+\delta \right)t}\left(k_{0}^{1-\alpha }-\frac{sa}{n+\delta }\right)\right]^{\frac{1}{1-\alpha }}
\]
\[
y (t) =ak^{\alpha }=a\left[\frac{sa}{n+\delta }+e^{-\left({1-\alpha }\right)\left(n+\delta \right)t}\left(k_{0}^{1-\alpha }-\frac{sa}{n+\delta }\right)\right]^{\frac{\alpha }{1-\alpha }}
\]
