<p>If <span class="math-container">$\Pi_t$</span> is gross inflation then indeed <span class="math-container">$(\Pi_t-1)^2$</span> 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.</p>
<p>To see this another way, use the &quot;trick&quot; of expressing inflation as the ratio of price levels and then log-linearize <span class="math-container">$P_t$</span> around <span class="math-container">$P_{t-1}$</span> as below:
<span class="math-container">$$
Y_t(\Pi_t - 1)^2 = Y_t(\frac{P_t}{P_{t-1}} - 1)^2 \approx Y_t\biggl(\frac{P_{t-1}(1+p_t)}{P_{t-1}} - 1\biggr)^2 = Y_t p_t^2 \approx 0
$$</span></p>

CC BY-SA 4.0
2023-02-08T21:47:14.777Z
2023-02-08T21:47:14.777Z
BrsG
<p>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 (<a href="https://www.researchgate.net/publication/228288770_Log-Linearizing_Around_the_Steady_State_A_Guide_with_Examples" rel="nofollow noreferrer">see here</a>) and I want to disregard all second order terms. <span class="math-container">$\bar{x}$</span> are steady-state values and <span class="math-container">$\hat{x_t}$</span> are percentage steady-state deviations.
This is the term in question:
<span class="math-container">\begin{align}
 Y_t(\Pi_t - 1)^2 = Y_t (\Pi_t^2 - 2\Pi_t +1)
\end{align}</span>
Now if I just try to log-linearize in the usual way, I end up with this:
<span class="math-container">\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}</span>
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 <span class="math-container">$(\Pi_t-1)^2$</span> is a second order term and close to zero for low inflation levels?</p>

<dsge><steady-state><log-linearization>
Log-linearizing a second order term around the steady-state
mindandfields
# Log-linearizing a second order term around the steady-state

(Asked by: username_0 on 2023-02-07)

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](https://www.researchgate.net/publication/228288770_Log-Linearizing_Around_the_Steady_State_A_Guide_with_Examples)) 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?

**Tags:** dsge, steady-state, log-linearization

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## 1 answers

### Answer by username_1 on 2023-02-08. Score: 3 | Accepted answer

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:
$$
Y\_t(\Pi\_t - 1)^2 = Y\_t(\frac{P\_t}{P\_{t-1}} - 1)^2 \approx Y\_t\biggl(\frac{P\_{t-1}(1+p\_t)}{P\_{t-1}} - 1\biggr)^2 = Y\_t p\_t^2 \approx 0
$$


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