AK Model
- Paul Romer and Robert Barro
- Assumes linearity
$$ Y = AK $$ is the production function where, - \(A\) technology - \(K\) capital, which can be - Physical - Human
There are no diminishing returns, marginal productivity of capital cannot be negative.
\[
\begin{align}
\dot{K} &= \underbrace{sY}_{\text{Grows > }\delta K} - \delta K \\
\dfrac{\dot{K}}{K} & = \dfrac{sY}{K} - \delta \\
\dfrac{\dot{K}}{K} & = \dfrac{sAK}{K} - \delta \\
\dfrac{\dot{K}}{K} & = sA - \delta \\
\end{align}
\]
\[
\begin{align}
Y & = AK \\
\ln Y & = \ln A + \ln K
\end{align}
\]
Assuming \(A\) is constant so \(\dfrac{d}{dt}A = 0\) and on differentiation of the above we get,
\[\dfrac{\dot{Y}}{Y} = \dfrac{\dot{K}}{K} = sA - \delta\]
Output is an \(\uparrow\) function of investment.
- Ideas (increasing returns to scale and monopolistic competition and due to patents)
- non-rivalrous
- public goods
- non-excludable
- For ideas, perfect competition not fit, because they incur losses. Because, they need to set \(MC = P\) and \(MC = 0\), so they need to price higher and these is no efficient pricing.