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!