independanteng

Answered

2022-10-31

Separating the log of a sum

I know there is no formula to separate the log of a sum, e.g. $\mathrm{log}(X+Y)$ into two parts, but are there any approximation rules that can allow me to achieve this objective?

${E}_{t}(1+{r}_{t+1}^{K})={E}_{t}\left[{\displaystyle \frac{\frac{1}{{X}_{t+1}}\alpha {A}_{0}\frac{{Y}_{t+1}}{{K}_{t+1}}+{Q}_{t+1}(1-\delta )}{{Q}_{t}}}\right]$

Suppose we ignore the expectations operator for the moment.

I know there is no formula to separate the log of a sum, e.g. $\mathrm{log}(X+Y)$ into two parts, but are there any approximation rules that can allow me to achieve this objective?

${E}_{t}(1+{r}_{t+1}^{K})={E}_{t}\left[{\displaystyle \frac{\frac{1}{{X}_{t+1}}\alpha {A}_{0}\frac{{Y}_{t+1}}{{K}_{t+1}}+{Q}_{t+1}(1-\delta )}{{Q}_{t}}}\right]$

Suppose we ignore the expectations operator for the moment.

Answer & Explanation

megagoalai

Expert

2022-11-01Added 22 answers

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

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