Conceptual question about the zero-inflation steady-state in DSGE-models

Question

As I understand, it is usually common practice, to choose DSGE models in such a way, that the steady-state inflation is freely choose-able and for simplifications it is than usually assumed to be zero.

My question is now, how is this usually achieved?

For a while now, I have been trying to replicate this paper. It appears to me, that the author does exactly that, he assumes a steady-state inflation of zero and log-linearizes the model around it. However, when I try to confirm this and solve the steady-state calculations, the model does not appear to have a freely choose-able steady-state inflation and in fact, it is different from zero. When I solve the steady-state for this model, I unsurprisingly get 12 equations and 12 variables and the model solves exactly. I assume this happens quite often, is there a trick to achieve a undetermined steady-state inflation?

As a second question, in the model above, the author did not specify the monetary policy in the beginning, but when he later summarized his log-linearized equations, he specified a simple taylor rule, of the form: \begin{align} \hat{R_t} = \phi_\Pi \hat{\pi_t} + \phi_y \hat{Y_t} \end{align} In order to calculate the steady-state of the model (before the log-linearizations), I assumed the nonlinear monetary policy equation to be: \begin{align} R_t = \Pi_t^{\phi_\pi} * Y_t^{\phi_y} \end{align} This made sense to me, since when log-linearizing this equation it becomes the equation specified by the author and I therefore thought, that it is the correct nonlinear counterpart of the monetary policy equation. But now I think this assumption may be my mistake and is why in my steady-state calculations, the inflation can not be zero.

Answer 1

In case someone ever has the same issue. I understand my problem now. When doing the backwards log-linearization, I already divided by the steady_state equation. But actually I shouldn't have done that. The nonlinear Taylor-Rule should look like this: \begin{align} \frac{R_t}{\bar{R}} = \bigg(\frac{\Pi_t}{\bar{\Pi}}\bigg)^{\phi_\pi} * \bigg(\frac{Y_t}{\bar{Y}}\bigg)^{\phi_y} \end{align} This way, the equation cancels in the steady-state and I can freely choose a steady-state inflation, as long as it is consistent with the real rate determined by preferences. This was not possible before, since the nominal rate was fixed by this equation.

And after log-linearizing, the steady-states cancel and we still achieve the regular Taylor Rule stated in the question.