I'm currently trying to retrace a log-linearization done in a paper. I want to log-linearize around the steady-state, as it is commonly done for DSGE models (see here) and I want to disregard all second order terms. $\bar{x}$ are steady-state values and $\hat{x_t}$ are percentage steady-state deviations. This is the term in question: \begin{align} Y_t(\Pi_t - 1)^2 = Y_t (\Pi_t^2 - 2\Pi_t +1) \end{align} Now if I just try to log-linearize in the usual way, I end up with this: \begin{align} \bar{Y}(1+\hat{Y_t})\biggl(\bar{\Pi}^2(1+2\hat{\Pi_t}) - 2\bar{\Pi}(1 + \hat{\Pi_t}) - 1\biggr) \end{align} So even after ignoring all second order terms, the term is still long and annoying, however, in the paper the whole term just disappears. Is there a better way to handle the log-linearization here and show how the term disappears? Or did the author just ignore the term from the get-go, because $(\Pi_t-1)^2$ is a second order term and close to zero for low inflation levels?
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If $\Pi_t$ is gross inflation then indeed $(\Pi_t-1)^2$ is a second-order term and is approximately zero. For example, for a reasonable quarterly steady state gross inflation rate of 1.005, the term in the brackets becomes very small when squared (0.000025) and even if the model is annual, it's still quite small.
To see this another way, use the "trick" of expressing inflation as the ratio of price levels and then log-linearize $P_t$ around $P_{t-1}$ as below: $$ Y_t(\Pi_t - 1)^2 = Y_t(\frac{P_t}{P_{t-1}} - 1)^2 \approx Y_t\biggl(\frac{P_{t-1}(1+p_t)}{P_{t-1}} - 1\biggr)^2 = Y_t p_t^2 \approx 0 $$