Different Steady State

In Woodford’s book “Interest and Price”,
Proposition 7.3 says that optimal ( long run ) inflation rate is negative if transactions frictions are considered.
The conclusion is derived using AS equation(1.1) and IS equation(1.15), which are liner approximations around the zero inflation steady state.

the question is:
How can we trust the conclusion which says the optimal rate is negative based on a system whose steady state is zero inflation rate?

nyone help me. Thanks.

Why should that be a problem? As long as the linear approximation is sufficiently close to the true functional form, the conclusion should be valid.

Thank you, dear jpfeifer.

“the linear approximation is sufficiently close to the true functional form”, this is true only near the zero inflation rate.
How can I trust this the linear approximation which brings a non-zero optimal inflation rate?

If your true function is close to linear, the approximation will be really good even if you are relatively far away from steady state. Of course, unless you are exactly in steady state, there will always be an approximation error. The good question is how big this error is.

You have to trust the papers that show that unless you have very strong non-linearities or really large shocks in your model the approximation errors are relatively small when using first order approximations. If you want to be 100% sure, you have to compute the Euler errors yourself.

I don’t know how far from the zero inflation steady state the optimal inflation rate is. But usually you get something like the Friedman rule, which is deflation of the size of the real interest rate. Usually, the real interest rate is set to about 1% per quarter, which implies optimal inflation of -1%. In almost all models this is close enough to steady state to deliver a really good approximation (consider that linearized models are often sufficient for business cycle analysis, e.g. Smets/Wouters, but the standard deviation of investment deviations from steady state is somewhere around 3-5 %).

Dear jpfeifer, Thank you very much!