When do workers versus capital owners share of income increase?
From the shares equation, we obtain:
$$
1 + \frac{rK}{wL} = \frac{1}{s_L}
\to tk = \frac{1 - s_L}{s_L}
$$
where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$.
then taking logs gives:
$$
\ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L)
$$
Then take the derivative of this expression with respect to $\ln(t)$:
$$
\begin{align*}
-\sigma &amp;= -1 - \dfrac{\dfrac{\partial s_L}{\partial \ln(t)}}{1 - s_L} - \gamma,\\
&amp;=-1 - \gamma - \frac{s_L}{1 - s_L} \gamma = -\left(\frac{\gamma}{1 - s_L} +1\right)
\end{align*}
$$
Where we defined $\gamma$ to be the elasticity of the labour revenue share with respect to the $r/w$ ratio.
Solving for $\gamma$ gives:
$$
\gamma = \left(\sigma-1\right)(1 - s_L)
$$
which means that $\gamma &gt; 0$ if $\sigma &lt; 1$. In other words, if the elasticity of subsititution is smaller than one, then the labour share increases when $r/w$ increases.