The AK model of economic growth is an endogenous growth model used in the theory of economic growth

Economic growth

In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...

, a subfield of modern macroeconomics

Macroeconomics

Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...

. Around 1980s it became progressively clearer that the standard neoclassical endogenous growth models were theoretically unsatisfactory as a tool to explore long run growth as these models predicts economy without technological change and eventually converge to a steady state with zero per capita growth as fundamental reason being diminishing return to capital and the key property of AK endogenous-growth model is the absence of diminishing returns to capital. In lieu of the diminishing returns to 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

Endogenous growth theory

Endogenous growth theory holds that economic growth is primarily the result of endogenous and not external force. In Endogenous growth theory investment in human capital, innovation and knowledge are significant contributors to economic growth. The theory also focus on positive externalities and...

.

Origin of the concept

In neoclassical growth model the economy is assumed to reach a steady state in which all the macroeconomic variable grow at a same rate and in the absence of technological progress, per capita growth of these macroeconomic variable will eventually seize. These kind of neo-classical prepositions have the resemblance with the philosophical contents in Ricardo and Mathus. The basic underline assumption of neo-classical philosophical is that diminishing returns to capital operates in the production process.
During mid-80s, a new beginning of growth theory was emerged by Romer (1986) where he tried to explain the growth process in a different manner. Thus the dissatisfaction out of neo-classical 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.

This equation shows a Cobb–Douglas function where Y represents the total production in an economy. A represents total factor productivity

Total factor productivity

In economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. If all inputs are accounted for, then total factor productivity can be taken as a measure of an economy’s long-term technological change or technological dynamism.If all inputs...

, K is capital, L is labor, and the parameter measures the output elasticity

Output elasticity

In economics, output elasticity is the percentage change of output divided by the percentage change of an input. It is sometimes called partial output elasticity to clarify that it refers to the change of only one input....

 of capital. For the special case in which , 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.
n = population growth rate

δ = depreciation

Depreciation

Depreciation refers to two very different but related concepts:# the decrease in value of assets , and# the allocation of the cost of assets to periods in which the assets are used ....

k = capital per worker

y = output/income per worker

L = labor force

s = saving rate
In an alternative form Y=AK, K embodies both physical capital and human capital.

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:
Y/L = A. K/L

i.e. y = A K

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

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:

k(t) = s. f(k) – nk

Hence, k(t) / k = (s. f(k) / k ) – n

But in the model ( f(k) / k ) = A

Thus, k(t) / k = s. A - n

Footnotes

Economic growth by Robert by J. Barro and Xavier Sala-i-Martin (2nd edition)

http://books.google.co.in/books/about/Economic_growth.html?id=jD3ASoSQJ-AC
http://mitpress.mit.edu/catalog/item/default.asp?tid=9923&ttype=2