# AK Model

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- Additional Contributors: None
Categories: CategoryModel CategoryMacro CategoryGrowth

Related Models: Solow_Growth_Model

Created: 2008-07-28

Suggested Citation: See license page.

## Introduction

In the 'spillover' models of economic growth, technological progress is an accidental 'by-product' of output by private profit-maximising firms. There are positive externalities to output, so although individual firms face constant returns to scale with diminishing returns to capital ($$\alpha<1$$), under some circumstances aggregate output is linear in capital. This means that output will not converge to a steady-state level, rather will continue to grow at a positive rate indefinitely. Changes to 'policy' parameters have both level and growth effects.

## Model

The economy consists of $$ j $$ identical firms with the production function $$ Y_j = \bar{A}K_j^{\alpha}L_j^{1-\alpha} $$ where $$ \bar{A} = A(\frac{K}{L})^{\mu} $$ . The parameter $$ \mu $$ measures the degree of 'knowledge spillover' from individual firms to the general level of technology in the industry, although individual firms take $$ \bar{A} $$ as given. In equilibrium all firms make the same enployment decisions so $$ Y = JY_j = \bar{A}K^{\alpha}L^{1-\alpha} = A(\frac{K}{L})^{\mu}K^{\alpha}L^{1-\alpha}$$ , which can be written in per-capita terms as $$ y = Ak^{\mu + \alpha} .

Capital is accumulated as in the fundamental Solow equation $$ \dot{K} = sY - \deltaK $$ which in per-capita terms is $$ \dot{k} = sAk^{\mu + \alpha} - (n+\delta)k $$ . This gives a growth rate of capital per-capita ($$\frac{K}{L}$$) of $$ \frac{\dot{k}}{k} = sAk^{\mu + \alpha - 1} - (n+\delta) $$ .

There is no technological change ($$ \frac{\dot{A}}{A} = 0 $$) , only an exogenously given constant level of knowledge $$ A $$ . This model shows that even without technological progress, the presence of externalities overcomes the constraint of diminishing returns at the individual firm level and allows for constant positive long-run growth.

## Solving

$$ \frac{\dot{y}}{y} = (\mu + \alpha) \frac{\dot{k}}{k} = (\mu + \alpha)[sAk^{\mu + \alpha - 1} - (n+\delta)] $$ :

If $$(\mu + \alpha) > 1 $$ then there is 'explosive' growth such that $$ \frac{\dot{y}}{y} $$ is increasing with $$ k $$ .

If $$(\mu + \alpha) < 1 $$ then the economy converges to a steady-state growth rate with $$ \frac{\dot{y}}{y} = 0 $$ . In this case 'knowledge spillovers' $$ \mu $$ have not been enough to overcome diminishing returns to capital, so the problem reverts to a version of the Solow model.

- If $$(\mu + \alpha) = 1 $$ then output is linear in capital and the growth equation reduces to $$ y = Ak $$ .

## Example: The AK Model

If $$(\mu + \alpha) = 1 $$ then $$ \frac{\dot{y}}{y} = \frac{\dot{k}}{k} = sA - (n+\delta) $$ , i.e. there is a constant growth rate, which is positive if $$ sA > (n+\delta) $$ . This means that it is possible to increase the growth rate through policy, for example changing the savings rate $$ s $$ .

This model does not predict convergence of any kind. The initial level of $$ k $$ or $$ y $$ does not affect the growth rate, and there is no steady-state level of output.