<p>We have that:
<span class="math-container">$$
L(s,H) = \frac{f(\theta)}{s + f(\theta)}H.
$$</span>
Take the derivative with respect to <span class="math-container">$\theta$</span>:
<span class="math-container">$$
\frac{\partial L}{\partial \theta} = \frac{f'(\theta)}{s + f(\theta)}H - \frac{f(\theta)}{(s + f(\theta))^2} f'(\theta) H
$$</span></p>
<p>Dividing by <span class="math-container">$L$</span> and mulitplying by <span class="math-container">$\theta$</span> gives:
<span class="math-container">$$
\begin{align*}
\epsilon_L &amp;= \frac{\partial L}{\partial \theta}\frac{\theta}{L} = \frac{f'(\theta)}{s + f(\theta)} H \frac{\theta (s + f(\theta)}{f(\theta)H} - \frac{f(\theta)}{(s + f(\theta))^2} f'(\theta) H \frac{\theta (s + f(\theta))}{f(\theta)H},\\
&amp;= \epsilon_f - \frac{f(\theta)}{s + f(\theta)} f'(\theta)\frac{\theta}{f(\theta)},\\
&amp;= \epsilon_f - \frac{f(\theta)}{s + f(\theta)} \epsilon_f
\end{align*}
$$</span></p>

CC BY-SA 4.0
2024-02-14T06:45:13.170Z
2024-02-14T06:45:13.170Z
tdm
<p>Labor supply is</p>
<p><span class="math-container">$ L(s,H) = \frac{f(\theta)}{s + f(\theta)} H $</span></p>
<p>Now the elasticity wrt to tightness <span class="math-container">$\theta$</span> is supposed to be</p>
<p><span class="math-container">$\epsilon_L= \epsilon_f - \frac{f(\theta)}{s + f(\theta)} \epsilon_f$</span></p>
<p>where <span class="math-container">$\epsilon_f$</span> is the elasticity of <span class="math-container">$f$</span> wrt to tightness. Why? If I take substitute in <span class="math-container">$ L(s,H)$</span> and take the derivative with the quotient rule I end up nowhere..</p>

<elasticity><search-and-matching>
elasticity of labor supply
Papayapap
# elasticity of labor supply

(Asked by: username_0 on 2024-02-13)

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..

**Tags:** elasticity, search-and-matching


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## 1 answers

### Answer by username_1 on 2024-02-14. Score: 4 | Accepted answer

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:
$$
\begin{align\*}
\epsilon\_L &= \frac{\partial L}{\partial \theta}\frac{\theta}{L} = \frac{f'(\theta)}{s + f(\theta)} H \frac{\theta (s + f(\theta)}{f(\theta)H} - \frac{f(\theta)}{(s + f(\theta))^2} f'(\theta) H \frac{\theta (s + f(\theta))}{f(\theta)H},\\
&= \epsilon\_f - \frac{f(\theta)}{s + f(\theta)} f'(\theta)\frac{\theta}{f(\theta)},\\
&= \epsilon\_f - \frac{f(\theta)}{s + f(\theta)} \epsilon\_f
\end{align\*}
$$


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