While going through the derivation of elasticity of substitution between capital and effective labour in economic materials for a Slow growth model, I found the following step there:
$\frac{\partial ln(K/AL)}{\partial ln(F_K/F_{AL})}$ = $\frac{\partial ln(K/F)}{\partial ln(F_K)}$,
when rewriting a formula for $\sigma$ (elasticity of substitution between capital and effective labour). Here $F$ is a production function. I was trying to reproduce that step, but wasn't successful yet... I tried to use the fact that $F$ is homothetic, but this fact alone didn't allow me to answer my question, i.e., I was trying to simplify both equations writing down those derivatives and doing a substitution $F=F_K\cdot K+F_{AL}\cdot AL$ for $F$.
Does anyone has other ideas? I think, the derivation was within a problem 1.10 of "Advanced Macroeconomics" from D. Romer.
