MRW

\[ Y = K^\alpha (AH)^{1-\alpha} \]

where,

• \(AH:\) skilled labor force (human capital with increasing returns to scale)
• \(A:\) technological progress, skill exogenously growing

In order to acquire skills, some time is foregone from work (like consumption is foregone (\(\implies\) savings) to buy capital)

\[ \underbrace{H}_{\text{Skilled Labor}} = e^{\phi q}L \]

where,

• \(q:\) hours spent in training
• \(L:\) raw unskilled labor

Let's try taking the intensive form

\[ \dfrac{Y}{L} = \dfrac{K^\alpha}{L^\alpha} \dfrac{(AH)^{1-\alpha}}{L^{-\alpha} \cdot L} \]

which gives us,

\[ y = k^\alpha (Ah)^{1-\alpha} \]

Dividing by \(Ah\) we get,

\[ \begin{gather*} \dfrac{y}{Ah} = k^\alpha (Ah)^{-\alpha} \\ \dfrac{y}{Ah} = \dfrac{k^\alpha}{(Ah)^{\alpha}} \\ \boxed{\dfrac{y}{Ah} = \left(\dfrac{k}{Ah}\right)^{\alpha}} \end{gather*} \]

This gives us,

\[ \tilde{y} = \tilde{k}^\alpha \]

Next,

\[ \tilde{k} = s\tilde{y} - (n+g+\delta)\tilde{k} \]

At steady state, \(\tilde{k} = 0\),

\[ s\tilde{y} = (n+g+\delta)\tilde{k} \]

Substituting \(\tilde{y} = \tilde{k}^\alpha\),

\[s\tilde{k}^\alpha = (n+g+\delta)\tilde{k}\]

With a little bit of algebraic manipulation,

\[ \tilde{k}^{(1-\alpha)} = \dfrac{s}{n+g+\delta} \]

Here, \(\dfrac{1}{1-\alpha}\) is the power of ?? so,

\[ \tilde{k} = \left(\dfrac{s}{n+g+\delta}\right)^{1/(1-\alpha)} \]

\[ \tilde{y_{t}} = \left(\dfrac{s}{n+g+\delta}\right)^{1/(1-\alpha)}A(t)h \]

This tells us why some countries continue to remain rich and others, poor.