Home > Macro, Solow Model > The Solow model and the production function

## The Solow model and the production function

I’ve presented the algebra of the Solow model in both the basic per capita case (without technological progress):

$k* = \frac{s}{(n +\delta)}y*$ and $y* = \frac{(n +\delta)}{s}k*$.

$\hat{k*} \approx \frac{s}{[\pi+n +\delta]}\hat{y*}$ and $\hat{y*} \approx \frac{[\pi+n +\delta]}{s}\hat{k*}$.

Now we can think of how this relates to a production function.

Typically the production function used to model output in the economy is of the form $Y=K^{\alpha}L^{1-\alpha}$. Here $\alpha$ is the share of capital in output and $1-\alpha$ is the share of labour in output. This particular form of production function is used to model the economy because it has constant returns to scale, which is fairly realistic (if you cloned an economy, it would probably produce the same amount of output, so by that logic if you double capital and labour in an economy, you would get double the amount of output).

If we want to use the small letter per-capita notation again, to show output and capital in ‘per worker’ terms, we can divide everything through by L: $\frac{Y}{L}=K^{\alpha}L^{-\alpha}$ so $y=\frac{K^{\alpha}}{L^{\alpha}} \Rightarrow y=k^{\alpha}$.

If $y=k^{\alpha}$ then $k=y^{\frac{1}{\alpha}}$

We can use this simple result, to express those earlier Solow equations in terms of the share of capital in output.

If in steady state, $k* = \frac{s}{(n +\delta)}y*$
then $k* = \frac{s}{(n +\delta)}k*^{\alpha} \Rightarrow k*^{1-\alpha} = \frac{s}{(n +\delta)} \Rightarrow k* = [\frac{s}{(n +\delta)}]^\frac{1}{1-\alpha}$.

And as $y=k^{\alpha}$
this means $y* = [\frac{s}{(n +\delta)}]^\frac{\alpha}{1-\alpha}$.

Similarly in the model with technological progress,

$k* \approx [\frac{s}{(\pi + n +\delta)}]^\frac{1}{1-\alpha}$
and
$y* \approx [\frac{s}{(\pi + n +\delta)}]^\frac{\alpha}{1-\alpha}$