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, $\eta_{\lambda, \theta}\cdot \eta_{\theta, J}$?


1 Answer 1


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 $$\eta_{\lambda, J}= \eta_{\lambda, \theta} \cdot\eta_{\theta, J} $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.