I have a dynamic equation,
$$ \frac{\dot{k}}{k} = s k^{\alpha – 1} + \delta + n$$
Where $\dot{k}/k$ is the capital growth rate as a function of savings $s$, capital $k$, capital depreciation rate $\delta$, and population growth rate $n$.
I have been asked to find the change in the growth rate as $k$ increases. This is of course
$$\frac{\partial \dot{k}/k}{\partial k} = (\alpha – 1) s k^{\alpha -2}$$
But what I want to find now is the change in growth rate as $k$ increases proportionately. This should be
$$\frac{\partial \dot{k}/k}{\partial \ln(k)} = ?$$
How do you calculate the partial derivative with respect to the logarithm of a variable? I'm sure the answer is simple, but my analytical calculus is pretty rusty.
Best Answer
The simplest way is via chain rule: $$ \dfrac{\partial \dot{k}/k}{\partial k} = \dfrac{\partial \dot{k}/k}{\partial \ln{k}}\dfrac{\partial \ln{k}}{\partial k} $$ ...from which you can move things around to get your quantity of interest, e.g. $$\dfrac{\partial \dot{k}/k}{\partial \ln{k}} = \dfrac{\partial \dot{k}/k}{\partial k}k = (\alpha-1)sk^{\alpha-1}$$ (since $(\ln{x})' = 1/x$).