Hello everyone

I am working on a model with positive inflation in the steady state in a non-linear form. The central bank’s target inflation rate is 3% however the mean of inflation in the data is around 4% so if I assume in the model that \Pi=1.0075 (i.e. 3% annual net rate) it would not be consistent with the mean of inflation in the data. Some time ago I read in the forum that inflation in the steady state does not have to be the same as the target inflation (\Pi^T), however, if I assume that \Pi\neq\Pi^T and I have a Taylor rule of the following form:

\frac{R_t}{R}=\left(\frac{R_{t-1}}{R}\right)^{\phi_R}{\left(\left(\frac{\Pi_t}{\Pi^T}\right)^{\phi_\pi}\left(\frac{Y_t}{Y}\right)^{\phi_y}\left({dS}_t\right)^{\phi_s}\right)}^{1-\phi_R}

then in the steady state I would have:

1=\left(\frac{\Pi}{\Pi^T}\right)^{\phi_\pi}

which obviously prevents the residuals of the equation from being zero in the steady state. Does this mean that I need to create a different parameter to replace R and adjust the residuals? Let’s call this new parameter R^T, so my taylor rule should look like this?

\frac{R_t}{R^T}=\left(\frac{R_{t-1}}{R^T}\right)^{\phi_R}{\left(\left(\frac{\Pi_t}{\Pi^T}\right)^{\phi_\pi}\left(\frac{Y_t}{Y}\right)^{\phi_y}\left({dS}_t\right)^{\phi_s}\right)}^{1-\phi_R}

Therefore, in the steady state, rearranging terms I would have:

\frac{R}{R^T}=\left(\frac{\Pi}{\Pi^T}\right)^{\phi_\pi} which leads to R^T=\left(\frac{\Pi^T}{\Pi}\right)^{\phi_\pi}R

I don’t know if this is correct and I don’t know what economic meaning R^T would have, but if it is not correct, how do you make \Pi\neq\Pi^T without affecting the steady state of the Taylor rule?

Thanks in advance