I'm currently trying to retrace a log-linearization done in this paper. I want to log-linearize around the steady-state, as it is commonly done for DSGE models (see here). $\bar{x_t}$ are steady-state values and $\hat{x_t}$ are percentage steady-state deviations.
I'm looking at the Phillips-Curve (p.7) in the paper, but to make it more readable I ignored some factors and summarized the constants with $C$.
The term that I want to log-linearize now is:
\begin{align} \Pi_t(\Pi_t - 1) = \bigg(\frac{C_{t+1}}{C_t}\bigg)^{-1} \frac{Y_{t+1}}{Y_t} \Pi_{t+1}(\Pi_{t+1} - 1) + w_t A_t - C \end{align}
The author achieves the very simplified result:
\begin{align} \hat{\Pi_t} = \hat{\Pi_{t+1}} + \hat{w_t} + \hat{A_t} \end{align}
However, if I try to log-linearize the model in the usual way, I first get:
\begin{align} \bar{\Pi}e^{\hat{\Pi_t}}(\bar{\Pi}e^{\hat{\Pi_t}} - 1) = \frac{e^\hat{-C_{t+1}}*e^\hat{Y_{t+1}}}{e^\hat{-C_{t}}*e^\hat{Y_{t}}} \bar{\Pi}e^{\hat{\Pi_{t+1}}}(\bar{\Pi}e^{\hat{\Pi_{t+1}}} - 1) + \bar{w}\bar{A}*e^{\hat{w_t}+\hat{A_t}} - C \end{align}
And after taking first order Taylor approximations and substracting the steady-state, the best I can do is this:
\begin{align} \hat{\Pi_t}*(2\bar{\Pi}^2 - \bar{\Pi}) = \bar{\Pi}^2*\bigg(\hat{Y_{t+1}} - \hat{Y_t} - (\hat{C_{t+1}} - \hat{C_t})\bigg) + \hat{\Pi_{t+1}}*(2\bar{\Pi}^2 - \bar{\Pi}) + \bar{w_t}\bar{A_t} * (\hat{w_t}+\hat{A_t}) \end{align}
Now I can see, that the term $\hat{Y_{t+1}} - \hat{Y_t} - (\hat{C_{t+1}} - \hat{C_t})$ could disappear, if the growth in output equals the growth in consumption, however, I don't understand how the author was able to get rid of all the remaining steady-state values. It almost looks like he divided by the steady-state equation, as you would normally do in multiplicative equations, but this is not possible here. Any ideas?
