# AK model

The AK model of economic growth is an endogenous growth model used in the theory of economic growth, a subfield of modern macroeconomics. In the 1980s it became progressively clearer that the standard neoclassical exogenous growth models were theoretically unsatisfactory as tools to explore long run growth, as these models predicted economies without technological change and thus they would eventually converge to a steady state, with zero per capita growth. A fundamental reason for this is the diminishing return of capital; the key property of AK endogenous-growth model is the absence of diminishing returns to capital. In lieu of the diminishing returns of capital implied by the usual parameterizations of a Cobb–Douglas production function, the AK model uses a linear model where output is a linear function of capital. Its appearance in most textbooks is to introduce endogenous growth theory.

## Origin of the concept

In neoclassical growth models the economy is assumed to reach a steady state in which all macroeconomic variables grow at the same rate and in the absence of technological progress, per capita growth of these macroeconomic variable will eventually cease. These kind of neoclassical prepositions have the resemblance with the philosophical contents in Ricardo and Malthus. The basic underlying assumption of neoclassical philosophical is that diminishing returns to capital operates in the production process.

During the mid-1980s a new beginning of growth theory launched by Paul Romer in 1986,[1] where he tried to explain the growth process in a different manner. Thus the dissatisfaction out of neoclassical model motivated to construct new growth theories where the key determination of growth theories are endogenous in the model as in these new theories, the long run growth is not determined by exogenous factors setting up endogenous growth theories.

The simplest version of endogenous model is AK models which assume constant exogenous saving rate and fixed level of technology. The stickiest assumption of this model is that production function does not include diminishing returns to capital. This means that with this strong assumption the model can lead to endogenous growth.

## Graphical representation of the 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, ${\displaystyle {\frac {k(t)}{k}}=s\cdot A-n}$