Labor supply is
$ L(s,H) = \frac{f(\theta)}{s + f(\theta)} H $
Now the elasticity wrt to tightness $\theta$ is supposed to be
$\epsilon_L= \epsilon_f - \frac{f(\theta)}{s + f(\theta)} \epsilon_f$
where $\epsilon_f$ is the elasticity of $f$ wrt to tightness. Why? If I take substitute in $ L(s,H)$ and take the derivative with the quotient rule I end up nowhere..
We have that: $$ L(s,H) = \frac{f(\theta)}{s + f(\theta)}H. $$ Take the derivative with respect to $\theta$: $$ \frac{\partial L}{\partial \theta} = \frac{f'(\theta)}{s + f(\theta)}H - \frac{f(\theta)}{(s + f(\theta))^2} f'(\theta) H $$
Dividing by $L$ and mulitplying by $\theta$ gives: $$ \begin{align*} \epsilon_L &= \frac{\partial L}{\partial \theta}\frac{\theta}{L} = \frac{f'(\theta)}{s + f(\theta)} H \frac{\theta (s + f(\theta)}{f(\theta)H} - \frac{f(\theta)}{(s + f(\theta))^2} f'(\theta) H \frac{\theta (s + f(\theta))}{f(\theta)H},\\ &= \epsilon_f - \frac{f(\theta)}{s + f(\theta)} f'(\theta)\frac{\theta}{f(\theta)},\\ &= \epsilon_f - \frac{f(\theta)}{s + f(\theta)} \epsilon_f \end{align*} $$
