I'm reading this book on macroeconomics.
When determining the nominal interest rate, the Central Bank has in mind its real interest rate goal. By the Fisher Equation: $i_t=r_t+\pi^{E_{CB}}_{t+1}$, since at time $t$, I would need to know the price level at time $t+1$ to know inflation for next period, we use an expected value for inflation. I put the superscript $^{E_{CB}}$ to denote that the expectancy is formed by the central bank.
In the usual Philips Curve, we have $\pi_{t+1}=\pi^{E_{SS}}_{t+1}+\alpha(y_{t+1}-y_e)$ I put the superscript $^{E_{SS}}$ to denote that the expectancy is formed by the Supply Side (wage and price setters, i.e., with all available public information).
Recently, from a talk with someone who worked at a central bank, I learned that there's some difference in amount/quality of information that it's publicly available to wage and price setters, and that which is available to the Central Bank.
In the book I'm reading, the authors state one possible way of escaping a deflation trap, even if in practice it will not always work, is that to have higher expectations so that the minimum threshold for the real interest rate is lower enough to increase output until equilibrium.
Now I was wondering if it was not possible for, in some situation, the CB publicly divulged information lead to workers and firms have higher expected inflation, while the CB would know those expectations to be a bit too optimistic, such that in reality that period real interest rate would be much lower to be enough to get out of deflation trap?
I think I must be making some errors in this reasoning.
Any help would be appreciated