Suppose you have the following nonlinear difference equation in terms of \( \Pi_{t} \) and \( R_{t} \) :
\( R_{t+1}^{\sigma-1} \beta^{\sigma} \frac{\Pi_{t}}{\Pi_{t+1}}=1-\Pi_{t} \)
At the steady state, \( R^{\sigma-1} \beta^{\sigma}=1-\Pi \) so that \( \Pi=1-R^{\sigma-1} \beta^{\sigma} \).
Loglinearize this equation around the steady state (denote the log-deviation of \( \Pi_{t} \) and \( R_{t} \) from their steady states by \( \pi_{t} \) and \( r_{t} \), respectively).
