We have output as a function of capital.
\(Y=f(K)\)
We also have capital dynamics.
\(\dot K=I-\delta K\)
\(I=S=sY\)
This gives us:
\(\dot K = sY-\delta K\)
The production function is:
\(Y=cK\)
This gives us:
\(\dot K=(sc-\delta )K\)
\(\dot Y=c\dot K\)
\(\dfrac{\dot Y}{Y}=c\dfrac{\dot K}{Y}\)
\(\dfrac{\dot Y}{Y}=c\dfrac{(sc-\delta )K}{cK}\)
\(\dfrac{\dot Y}{Y}=sc-\delta\)
Per capita income is:
\(y=\dfrac{Y}{L}\)
\(k=\dfrac{K}{L}\)
As with the Harrod-Domar model we have output as a function of capital:
\(Y=f(K)\)
Capital dynamics:
\(\dot K=I-\delta K\)
\(I=S=sY\)
This gives us:
\(\dot K = sY-\delta K\)
The production function of the Harrod-Domar model is:
\(Y=cK\)
And long-term growth of:
\(\dfrac{\dot Y}{Y}=sc-\delta\)
We use a new production function:
\(Y=K^\alpha (AL)^{1-\alpha }\)
We add dynamics for technology and labour.
\(A_t=A_0e^{gt}\)
\(L_t=L_0e^{nt}\)
\(Y_t=K_t^\alpha (A_0e^{gt}L_0e^{nt})^{1-\alpha }\)
\(k_t=\dfrac{K_t}{A_tL_t}\)
\(Y_t=\dfrac{Y_t}{A_tL_t}\)
The dynamics of effective capital is:
\(\dot k_t=sk_t^{\alpha }-(n+\delta +g)k_t\)
In equilibrium effective capital is stable.
\(\dot k_t^*=sk_t^{\alpha }-(n+\delta +g)k_t^8=0\)
\(sk_t^{*\alpha }=(n+\delta +g)k_t^8\)
\(k_t^*=\biggr (\dfrac{s}{n+g+\delta }\biggr )^{\dfrac{1}{1-\alpha }}\)
We add human capital to the Solow-Swan model.
The Golden Rule savings rate is the rate which maximises long term consumption per capita.
If the savings rate is \(0\) there is no capital and no income. If the savings rate is \(1\) then then there is no consumption.
This is based on the Solow-Swan model, with an endogeneous savings rate.
As with the Harrod-Domar model we have output as a function of capital:
\(Y=f(K)\)
Capital dynamics:
\(\dot K=I-\delta K\)
\(I=S=sY\)
This gives us:
\(\dot K = sY-\delta K\)
In the Solow-Swan model the production function was:
\(Y=K^\alpha (AL)^{1-\alpha }\)
In the Harrod-Domar model the production function was:
\(Y=cK\)
In the Solow-Swan model we also added population and technology growth
In the AK model the production function is:
\(Y=AK\)
We keep population growth from the Solow-Swan model.
\(\dot K = sAK-\delta K\)
\(\dot K = (sA-\delta )K\)
\(k=\dfrac{K}{L}\)
\(\dot k =\dfrac{\dot K}{L}-\dot L\dfrac{K}{L^2}\)
\(\dot k =\dfrac{(sA-\delta )K}{L}-\dot L\dfrac{K}{L^2}\)
\(\dot k =(sA-\delta )k-k\dfrac{\dot L}{L}\)
\(\dot k =(sA-\delta -n)k\)
\(\dfrac{\dot k}{k} =sA-\delta -n\)