Steady state inflation rate and nonzero inflation target

Dear Johannes,
First thank you very much for your previous guidance, I am grateful.
I have a question about relationship between steady state inflation and inflation target when monetary policy rule is inflation targeted and inflation target is nonzero (e.g. most central banks have inflation target 2%),
I think theoretically, in the steady state, inflation rate is zero, in my DSGE model, inflation is PAI=ln(Price/Price(-1)), hence, in steady state, Price=Price(-1), inflation=0.

However, I have used Taylor type interest rate feedback rule for monetary policy
R-steady_state( R )=rho * (R(-1)-steady_state( R ))+(1-rho) * thetay * (Y-steady_state(Y))+(1-rho) * thetapai * (PAI-Inflation_target)+(1-rho) * thetasg * (Stock_Growth-steady_state(Stock_Growth))+epsR
R is interest rate, Y is output, PAI is inflation, epsR is monetary policy shock, I treat inflation_target as a parameter to be estimated using Bayesian method. Initial value and prior mean of inflation_target is 2%.

I find a problem about what should steady state inflation be when inflation target is 2%?

Scenario 1, if I put steady state inflation PAI to 0 and inflation_target is nonzero,e.g.2%, then I find Taylor type interest feedback rule becomes inequality, and the program broke down.

Scenario 2. if I put steady state inflation PAI to 2%. then I find Taylor type interest feedback rule is satisfied. however, inflation is PAI=ln(Price/Price(-1)=0.02, it means that Price does not have a steady state,

Thank you very much and look forward to hearing from you.
Jesse
PhD Candidate

Typical New Keynesian models feature infinitely many steady states for prices. That is why we usually set up the model in inflation rates. The steady state of the inflation rate is also not endogenously determined, but follows from the inflation target of the central bank. If you say the central bank targets 2%, then that is also the steady state inflation rate. The question then becomes how to deal with the non-zero steady state inflation. One way is to have indexing (e.g. Smets/Wouter or Christiano/Eichenbaum/Evans). The other is to simply have non-indexed trend inflation, which then also affects determinacy, welfare, etc. See the work of Guido Ascari and https://github.com/JohannesPfeifer/DSGE_mod/blob/master/Ascari_Sbordone_2014/Ascari_Sbordone_2014.mod

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