<p>But it is, there is just a simple oversight in your calculations:
We can indeed write identically</p>

<p>$$\eta_{\lambda, J} = \frac{\partial \lambda}{\partial \theta}\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda} \\
\implies \eta_{\lambda, J}= \left\{\frac{\partial \lambda}{\partial \theta} \frac{\theta}{\lambda}\right\}\left\{\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda}\right\} \frac{\lambda}{\theta} $$</p>

<p>But the term in the second curly brackets is <em>not</em> equal to $\eta_{\theta, J}$.
We have to re-arrange to get</p>

<p>$$\eta_{\lambda, J}= \left\{\frac{\partial \lambda}{\partial \theta} \frac{\theta}{\lambda}\right\}\cdot \left\{\frac{\partial \theta}{\partial J} \cdot \frac{J}{\theta}\right\} \frac{\lambda}{\lambda} $$</p>

<p>The fraction in the end now cancels and so 
$$\eta_{\lambda, J}= \eta_{\lambda, \theta} \cdot\eta_{\theta, J} $$</p>

CC BY-SA 3.0
2015-03-07T02:38:14.583Z
2015-03-07T02:38:14.583Z
Alecos Papadopoulos
<p>Consider the following composite function</p>

<p>$$ \lambda(\theta(J)) $$</p>

<p>If you want to think about it in more economic terms, think about it how values of firms $J$ affects market tightness $\theta$, and through that, job finding rates $\lambda$.</p>

<p>Let $\eta_{a,b}$ denote the elasticity of $a$ w.r.t. $b$. Then we have, under some abuse of notation,</p>

<p>$$ \eta_{\lambda, J} = \frac{\partial \lambda}{\partial \theta}\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda} \\
\eta_{\lambda, J}= \frac{\partial \lambda}{\partial \theta} \frac{\theta}{\lambda}\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda} \frac{\lambda}{\theta} \\
\eta_{\lambda, J}= \eta_{\lambda, \theta} \eta_{\theta, J} \frac{\lambda}{\theta}
$$</p>

<p>Could someone give me an intuition for $\lambda / \theta$ in the last expression? Why is the total elasticity not just the chain of elasticities, $\eta_{\lambda, \theta}\cdot \eta_{\theta, J}$?</p>

<elasticity>
Composite Elasticities
FooBar
# Composite Elasticities

(Asked by: username_0 on 2015-03-06)

Consider the following composite function

$$ \lambda(\theta(J)) $$

If you want to think about it in more economic terms, think about it how values of firms $J$ affects market tightness $\theta$, and through that, job finding rates $\lambda$.

Let $\eta\_{a,b}$ denote the elasticity of $a$ w.r.t. $b$. Then we have, under some abuse of notation,

$$ \eta\_{\lambda, J} = \frac{\partial \lambda}{\partial \theta}\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda} \\
\eta\_{\lambda, J}= \frac{\partial \lambda}{\partial \theta} \frac{\theta}{\lambda}\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda} \frac{\lambda}{\theta} \\
\eta\_{\lambda, J}= \eta\_{\lambda, \theta} \eta\_{\theta, J} \frac{\lambda}{\theta}
$$

Could someone give me an intuition for $\lambda / \theta$ in the last expression? Why is the total elasticity not just the chain of elasticities, $\eta\_{\lambda, \theta}\cdot \eta\_{\theta, J}$?

**Tags:** elasticity

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## 1 answers

### Answer by username_1 on 2015-03-07. Score: 4 | Accepted answer

But it is, there is just a simple oversight in your calculations:
We can indeed write identically

$$\eta\_{\lambda, J} = \frac{\partial \lambda}{\partial \theta}\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda} \\
\implies \eta\_{\lambda, J}= \left\{\frac{\partial \lambda}{\partial \theta} \frac{\theta}{\lambda}\right\}\left\{\frac{\partial \theta}{\partial J} \cdot \frac{J}{\lambda}\right\} \frac{\lambda}{\theta} $$

But the term in the second curly brackets is *not* equal to $\eta\_{\theta, J}$.
We have to re-arrange to get

$$\eta\_{\lambda, J}= \left\{\frac{\partial \lambda}{\partial \theta} \frac{\theta}{\lambda}\right\}\cdot \left\{\frac{\partial \theta}{\partial J} \cdot \frac{J}{\theta}\right\} \frac{\lambda}{\lambda} $$

The fraction in the end now cancels and so
$$\eta\_{\lambda, J}= \eta\_{\lambda, \theta} \cdot\eta\_{\theta, J} $$


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