independanteng

2022-10-31

Separating the log of a sum
I know there is no formula to separate the log of a sum, e.g. $\mathrm{log}\left(X+Y\right)$ into two parts, but are there any approximation rules that can allow me to achieve this objective?
${E}_{t}\left(1+{r}_{t+1}^{K}\right)={E}_{t}\left[\frac{\frac{1}{{X}_{t+1}}\alpha {A}_{0}\frac{{Y}_{t+1}}{{K}_{t+1}}+{Q}_{t+1}\left(1-\delta \right)}{{Q}_{t}}\right]$
Suppose we ignore the expectations operator for the moment.

megagoalai

Expert

$\mathrm{log}\left(X+Y\right)=\mathrm{log}\left(X\right)+\mathrm{log}\left(1+Y/X\right)$, so if either $X$ is small compared to $Y$ or vice versa then you can approximate with a Taylor approximation.