# Steady state analysis using (nearly) permanent shocks?

Dear All,

I have a doubt. Suppose I have a linearized, calibrated model in which the steady state values of, say, output is fixed.

If I run an IRF to a technology shock with a 0.99 AR coefficient, I will see output rising to some positive percentage deviation from the steady state, and stay there for “a long time”.

Would that be a rough approximation of the new steady state which the model would have achieved, if it was non-linear and after a permanent change to steady state productivity?

Thanks!

More or less yes. For similar approach see for example: “Optimal moneatry policy response to distortionary tax changes” by Lubik and Lemke

Thank you, that is useful.

It is not as easy. As a rough approximation that is true. However, you do not know the transition behavior and the speed of adjustment to the “new steady state”. It could be that the first period after the impulse is actually closest to the new steady state or after a few periods. So only if there is not much movement between the periods after the shock and the subsequent ones is the response after some time representative of the “new steady state”.

Thank you jpfeifer, also that must be true.

But in such a case, then, what about inflation? Imagine you have set a zero inflation steady state, but after the “nearly-permanent” shock you have the endogenous variables, including inflation, jumping to (a neighborhood of) the new steady state.

Now suppose inflation stays different from zero. How would you interpret that?

Which model do you have in mind. In most models, inflation does not inherit unit roots.

Let me explain.

For example, in the contest of a NKPC, any nearly-permanent shock affecting marginal costs will make inflation inherit that shock, i.e. being above or below its steady state level.

Now, if the shock was truly permanent, marginal costs (but also all other endogenous variables) would reach their new steady state level. At the new steady state, inflation must be the same as before the shock (because that steady state inflation is determined outside the model).

But since the shock is only “nearly-permanent” (e.g. AR coeff. 0.99), this means that inflation will be above or below its ss level for a long time (so inheriting the persistence of the shock). In this sense, as for the other variables, one could be tempted to talk about “new steady state inflation”, but that’s a parameter… Do you see what I mean?

Think about what you are trying to achieve. You want to approximate a case with a unit root by a limit argument where you are close to a unit root. If you now consider the case of an exact unit root, you know that inflation won’t inherit this unit root and thus will not have a new steady state. Thus, the inflation deviation in the near unit root case should be still interpreted as a deviations from the old unchanged steady state.