Gali (2015) Chapter 8 Symmetric Steady State In Chapter 8 (small open economy) of Gali's 2015 version of Monetary Policy, Inflation, and the Business Cycle, he often performs log linearisations around a symmetric steady state like below:



My questions are:
1. What exactly is a symmetric steady state,
2. How do we know that it exists
3. How do we know that $P_H = P_F = P$ in steady state? (I know from S=1, that I can infer that $P_H = P_F$, but I'm confused about whether S=1 is the cause for $P_H = P_F$ or a consequence of $P_H = P_F$).

Thank you very much for your time in advance!
The steady state is &quot;symmetric&quot; in the sense that terms of trade with all other countries are equal to $1$, with respect to the home country. It would be non-symmetric if for some of them terms of trade were not equal to unity.

Is there anything in the model that precludes this steady state from existing? Usually not. Then it is feasible, i.e. it exists. Whether it is an attractive, stable steady state, is another matter.

$S$ is the ratio of two price levels. Logically then, it is a derivative measure of these two, so the price levels are the underlying forces.