# elasticity of labor supply

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*}