So, what's the basic idea? First, begin with the fact that monetary and fiscal policy are inextricably linked via a consolidated government budget constraint. This implies that monetary policy will have fiscal consequences. In particular, interest-rate policy affects the interest expense associated with rolling over any given amount of government debt. The question is how the fiscal authority intends to finance interest expense. There are two basic ways it can to do this: (1) use primary surplus (increase taxes and/or cut spending); (2) issue debt. The first option is associated with what economists call a Ricardian fiscal policy; the second option is associated with a Non-Ricardian fiscal policy.
A central bank has no control over (1) or (2); these are determined by the tax and spend decisions made by the fiscal authority. Suppose the fiscal authority chooses (2). If so, then what is accomplished by increasing the policy rate? Ceteris paribus, it increases the rate at which nominal debt is issued. This is arithmetic. If nominal debt is money (and it is), then this must lead to higher inflation, not lower inflation.
The corollary here is that a central bank has no unilateral control over the long-run rate of inflation. A central bank may be permitted to choose a long-run inflation target, but only with the blessing of the fiscal authority. Canadians know this: the Bank of Canada and the Government of Canada meet every five years to review their joint inflation-control agreement. In other jurisdictions, the central bank simply assumes that fiscal policy will be conducted in a "responsible" manner (i.e., a manner that will not un-anchor long-run inflation expectations).
If the primary deficit is managed in a manner to anchor long-run inflation, then the central bank is left free to use interest rate policy for the purpose of stabilizing shocks to aggregate demand. To stabilize the inflation rate around target requires, in this context, that the central bank raises its policy rate aggressively against above-target inflation. (Hopefully, even the threat of such a response keeps inflation close to target. This is the so-called Taylor principle.)
The question monetary policymakers might want to mull over these days is whether this standard policy prescription is appropriate in an environment where the traditional fiscal support for inflation-targeting seems to be waning? I want to be clear here: I am not saying this is happening today or that it will happen in the future. I'm saying that it might happen and that if it does, monetary policy makers should have a contingency plan in place. What should this contingency plan look like?
So, to take a concrete example, suppose that the tax and spend decisions coming from Congress imply an elevated primary deficit for the foreseeable future. Perhaps there's been a "regime change" in thinking that transcends political parties so, no matter who controls Congress, the expectation is for elevated primary deficits for as long as we can see. (I am not suggesting this is good or bad, I'm just saying suppose.)
Next, suppose the economy is humming along at or near what anyone would call "full employment." And then suppose inflation rises to 3, 4, 5% or higher and stays there with no sign of ever wanting to return to the Fed's official 2% long-run inflation target. What should the FOMC do in this hypothetical scenario? (Again, consider this as the type of thought-experiment that is necessary to form a contingency plan -- I do not mean to suggest that this scenario is likely, only that it is possible--and not in a Dumb and Dumber way).
The monetary policy advice coming from a model like Sargent and Wallace (1981) might suggest something like this: For as long as Congress remains in a regime of high primary deficits,
 Keep policy rate low, or even lower it, if possible; and
 Announce a temporarily higher inflation target (consistent with the new fiscal regime).
Recommendation  comes from Unpleasant Monetarist Arithmetic. Increasing the interest rate in this fiscal regime will only lead to higher inflation. Lowering the interest rate has the opposite effect. If recommendation  is not adopted, the monetary authority would have to explain (after every meeting) why it is missing its 2% inflation target. They might, of course, just say it's "temporary," but this would wear thin after a few years.
This advice is based on the assumption that everyone knows there's been a change in fiscal regime and that it will be persistent. What if no one is really sure of regime change or, if regime change, how long it might last?
The prudent thing to do in this more realistic case is to hedge your bets. In terms of , one might recommend raising the policy rate, but not by as much as would normally be done given the observed inflationary pressure. In terms of , one could probably get away with maintaining the long-run inflation target at 2% and legitimately explaining away deviations from target as "transitory."
Probably the last thing monetary policy should do under these circumstances is to raise the policy rate aggressively against inflationary pressure (as recommended by the Taylor principle). The Taylor rule works fine under a Ricardian fiscal policy. But it may backfire un a Non-Ricardian fiscal policy--this was the whole point of Sargent and Wallace (1981).
Tightening monetary policy might have the effect of bringing inflation down temporarily (this is consistent with the Sargent and Wallace model). But in reality (and in some models, like here and here), this would come at the cost of economic recession.
I can see no rationale for creating a recession to bring inflation down temporarily. But there may be a political-economy rationale for the threat of such a policy. That is, a Congress that does not trust future Congresses may want to create an independent (but accountable) central bank to pursue a low-inflation mandate and to do whatever it can with interest rate policy to achieve that mandate, even at the cost of recession. Future Congresses are in this manner obliged to behave in a Ricardian manner (so only temporary deficits permitted), which has the effect of anchoring long-run inflation.
Well, maybe. But I can't help but think of Dr. Strangelove here.
PS. I recently gave a talk on this that you can view here beginning at the 3 hour 12 minute mark.