Composite Elasticities

Question

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

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