I am evaluating simple and implementable monetary rules in the style of SGU, but for an open economy with incomplete exchange rate pass-through. The rules are evaluated with respect to the conditional welfare (conditional on the non-stochastic steady-state), and to compare welfare losses through consumption equivalent (CE), I´m using Ramsey welfare as the reference regime but unlike most papers, the Ramsey welfare is conditional on the deterministic (non-Ramsey) steady state. To do this I used the histval command following the recommendations of the forum.

When I calculate the CE, practically all the values of the different rules oscillate between 0.2 and 0.3, for example the CE for two rules is

and so for most rules, i.e, the 0.2 is kept and the CE only changes up to the third decimal. If I use an optimized rule as a reference instead of ramsey, I get more “usual” values, e.g. for the same rules but taking a simple optimized rule as a reference, the CE would be:

That is, when I use the Ramsey regime as a reference, all rules typically have a “base” welfare loss of 0.2%. Is this because I used Ramsey welfare conditional on the deterministic steady state instead of everything being conditional on the Ramsey steady state as in SGU (2006)(expanded version)?

I don’t know if it’s for this reason or if I made a mistake, I attach a mod file with the model where I calculate the CE and antoher one with the Ramsey model and the histval block

I am not sure I understand the issue. What you report seems to suggest that Ramsey is simply much better than even the best rule. Shifting the baseline by 0.2% when going from Ramsey to the best rule explains the numbers. The big question is why Ramsey is quite a lot better. Is it intended that the Ramsey solution has a unit root?

Yes, that is indeed the question, why Ramsey is much better than even the optimized rule, when in the literature the optimized rules are usually not that different from Ramsey. I hadn’t noticed the unit root problem in Ramsey’s model, but it seems strange to me since the model is the same as in the Calib_model mode file, which has no unit root. The only changes I made to the Ramsey file was to remove the Taylor rule, and make the steady state conditional on R (pretty much the only change was \Pi=R*\beta).

Any suggestion why only with those changes the Ramsey model exhibits unit root? (There was a bug in the constraint, I had R>0 and it was R>1, because is a gross interest rate but that doesn’t solve the problem of unit root)

I see, but the Fisher equation in steady state will typically introduce such a unit root, because inflation and nominal interest rate will move one for one.

Are you suggesting that the unit root could be expected and that it does not indicate something wrong with the model? Because I can’t get THEORETICAL MOMENTS from the different rates of inflations (which are all linked to \Pi=\beta*R) or the interest rate, is that normal?

Another weird thing I noticed is that the unconditional welfare value is too large (e.g. 3629693591.18639660)

You can verify easily that the steady state is not unique. Any initial value for R results in a steady state. Subsequently, the unit root makes nominal objects non-stationary. This explains the NaN moments and also the weird unconditional welfare.

I see, thanks a lot for you help, and sorry for bothering you with so many questions.

What is the usual way of proceeding in such cases? Is the Ramsey model still valid to take conditional welfare as a reference to calculate the CE?

Would that explain the big difference in welfare between the optimized rules and the Ramsey policy? That is the main problem that persists no matter what type of optimized rule I choose, the closest I have found has an EC of 0.09%, perhaps it has to do with the steady state that I choose from the many that the Ramsey model has, but the CE results look quite weird
Cheers

I think I found the reason for the unit root in Ramsey’s model, I assumed full indexation, that seems to generate the unit root problem, as soon as I set the parameters to assume partial indexation the unit root problem disappears.

The problem is that to do welfare analysis, I previously made estimates with real data assuming full indexation, in that case, is it better to take some other reference regime to calculate the CE? e.g. inflation targeting (\Pi_t=1 for all t) given the fact that this regime usually gives a level of welfare very similar to that of Ramsey planner?

Yes, I tried to estimate with partial indexation but there are 4 sectors in each one there are price rigidities, therefore there are 4 indexing parameters. With the values of the estimated parameters the model remains undetermined, also as the goal of the model is to evaluate monetary policy rules, it becomes almost impossible to test different rules due to Blanchard-Khan conditions, for that reason I had to assume full indexation

What do you mean by this?

my inflation in the steady state is 1.0075, but to calculate the consumption equivalent I took as a reference regime the conditional welfare when instead of the Taylor rule I have \Pi=1 which is the same welfare if I impose \Pi=1.0075 fo all t as a rule of policy, given the full indexation.

What I am saying is: you probably have a rule that hard-codes that it reacts to deviations in inflation from the steady state, where the steady state was preset at 1.0075. But now in Ramsey due to full indexation you allow steady state inflation to be anything.