Taylor rule in flexible price version of (MIU) NK Model

Hello everyone!
I have an NK model whose flexible price counterpart I am using to keep track of the output gap.
The way I compute the flexible price model is by just setting the price updaters proportion to 1 (theta=0), and that works just fine (e.g. Hours worked are up in the flex. model as opposed to down in the NK model following a TFP shock, as I think it should be for widely used combinations of param. values).
My question is, since the NK model has a Taylor rule, is keeping this rule just as is in the flex. price model okay? Cuz the thing is that with MIU utility, changes in the money supply do affect the potential of the economy as well, so does it make sense (or is it common knowledge in the literature) that the central bank keep the same objectives or (Taylor) rule even when prices are flexibles? (since, again, I think the classical dichotomy is broken by the MIU assumption). Or should I get rid of the Taylor rule but then end up with not quite the same (potential) model anymore?

I guess that depends on the specification of money in the utility function. If it is additively separable, then monetary neutrality should still hold for the rest of the economy.

Thank you for your reply Prof jpfeifer.
I was just afraid the MIU assumption might make central bank’s policy not neutral in the flexible price model, which could lead to unwanted effects from the monetary authority in the said version of the model

You should be able to test this. A monetary policy shock in the Taylor rule should not move any real variables.

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