AK model

The AK model production function is a special case of a Cobb–Douglas function with constant returns to scale.

${\displaystyle Y=AK^{a}L^{1-a}\,}$

This equation shows a Cobb–Douglas function where Y represents the total production in an economy. A represents total factor productivity, K is capital, L is labor, and the parameter ${\displaystyle a}$ measures the output elasticity of capital. For the special case in which ${\displaystyle a=1}$, the production function becomes linear in capital and does not have the property of decreasing returns to scale in the capital stock, which would prevail for any other value of the capital intensity between 0 and 1.

${\displaystyle n}$ = population growth rate
${\displaystyle \delta \ }$ = depreciation
${\displaystyle k}$ = capital per worker
${\displaystyle y}$ = output/income per worker
${\displaystyle L}$ = labor force
${\displaystyle s}$ = saving rate

In an alternative form ${\displaystyle Y=AK}$, ${\displaystyle K}$ embodies both physical capital and human capital.

${\displaystyle Y=AK\,}$

In the above equation A is the level of technology which is positive constant and K represents volume of capital. Hence, output per capita is:

${\displaystyle {\frac {Y}{L}}=A\cdot {\frac {K}{L}}}$ i.e. ${\displaystyle y=Ak}$

The model implicitly assumes that the average product of capital is equal to marginal product of capital which is equivalent to:

${\displaystyle A>0}$

The model again assumes that labor force is growing at a constant rate ‘n’ and there is no depreciation of capital. (δ = 0 ) In this case, the basic differential equation of neo-classical growth model would be:

${\displaystyle k(t)=s\cdot f(k)-nk}$

Hence, ${\displaystyle {\frac {k(t)}{k}}=s\cdot {\frac {f(k)}{k}}-n}$

But in the model ${\displaystyle {\frac {f(k)}{k}}=A}$

Thus, ${\displaystyle {\frac {k(t)}{k}}=s\cdot A-n}$

To avoid the contradictions, Russian economist Vladimir Pokrovskii proposed to write the production function in the united form

${\displaystyle Y={\begin{cases}\xi K,&\xi >0\\Y_{0}{\frac {L}{L_{0}}}\left({\frac {L_{0}}{L}}{\frac {P}{P_{0}}}\right)^{\alpha },&0<\alpha <1\end{cases}}}$

where ${\displaystyle P}$ is a capital severce; ${\displaystyle Y_{0}}$, ${\displaystyle L_{0}}$ and ${\displaystyle P_{0}}$ correspond to output, labour and substitutive work in the base year. This form of the theory explains growth as a consequence of the dynamics of the production factors, without any arbitrary parameters, which makes it possible to reproduce historical rates of economic growth with considerable precision.[2][3][4]

• Economic growth
• Human capital
• Harrod–Domar model

• Acemoglu, Daron (2009). "First-Generation Models of Endogenous Growth". Introduction to Modern Economic Growth. Princeton: Princeton University Press. pp. 387–407. ISBN 978-0-691-13292-1.
• Barro, Robert J.; Sala-i-Martin, Xavier (2004). "One-Sector Models of Endogenous Growth". Economic Growth (Second ed.). London: MIT Press. pp. 205–237. ISBN 0-262-02553-1.