How do I evaluate a partial derivative that is expressed as $$ \frac{\partial \ln \gamma}{\partial \ln x}, $$ where $\gamma$ is a function of $x$?
Is this expression simply short-hand for $$ \frac{x}{\gamma}\frac{\partial \gamma}{\partial x}, $$ Or is there more to it?
On
Yes, it is true. Consider the function $\gamma=x^{\alpha}, x>0$
We have $\frac{x}{\gamma}\frac{\partial \gamma}{\partial x}=\frac{x}{x^{\alpha}}\cdot \alpha\cdot x^{\alpha-1}=\alpha$
And $\ln(\gamma)=\alpha\cdot \ln\left(x\right)$ And therefore $\frac{\partial\ln(\gamma)}{\partial\ln\left(x\right)}=\alpha$
That are two ways to express the elasticity of a function.
The only difference is that $\frac{x}{\gamma}\frac{\partial\gamma}{\partial x}$ doesn't require $x,\,\gamma>0$.