I don't give the Fiscal Theory of the price level much credence.
Beckworth gives an equation where the real value of the monetary base plus the other portions of the national debt depend on the expected present value of government surpluses forever.
That requires that people must expect that an existing monetary regime last forever. Well, people might expect that, but it is not exactly rational.
No doubt I am biased because I favor a shift in monetary regime that would make it independent of expected future government budget surpluses. Cochrane's approach is that the price level today depends on the assumption that there is no chance that my preferred monetary reform is implemented ever. I am sad.
Anyway, I still don't believe it. Consider Beckworth's quote from Cochrane here:
The fiscal equation affects prices in an intuitive way. If people start to think surpluses will not be sufficient to pay off the debt, they try to unload government debt now, buying other assets or goods and services. This is just “aggregate demand.”
The problem with this analysis is that when people unload government debt now, the price of government debt falls and its yield increases. This tends to clear the market for government debt without any change in the price level.
But what about the monetary base? If people start to "unload" that, spending it on other assets, goods, or services, then the result is inflationary. But if the monetary regime adjust the quantity of base money with the demand to hold it, then any decrease in the demand for the monetary base simply results in a lower quantity at a constant "price." The price level remains the same.
If, as is conventional, the central bank sells government bonds, then this reinforces the tendency of government bond prices to fall and their yields to rise.
If we consolidate the balance sheet of the central bank and the government, what is happening is that less of the national debt is being financed by monetary liabilities and more by interest bearing debt. The price level remains stable and the interest rate on government bonds increases. Aggregate demand and the price level remain the same. Paying interest on base money would also help maintain the demand for it.
This process breaks down when the price of government bonds falls to zero or else demand to hold the monetary base falls to zero at the current price level.
A zero demand for base money doesn't mean that an inflation or nominal GDP target cannot be maintained, but it does mean that adjusting the quantity of base money according to the demand for it conditional on the target for the price level, inflation, or nominal GDP won't work. You are in a cashless payments system world.
Implicit in the Fiscal Theory of the Price Level is that money is held as an investment vehicle--it is just like government bonds. Since there is really no role for a medium of exchange in a general equilibrium model, then economists can't say anything about it, right? Further, if we calculate the price level in terms of interest bearing bonds, then a lower price of bonds is a higher price level. In a general equilibrium framework, why not? One numeraire serves as well as any other.
Anyway, suppose at some future date, a central bank owns lots of government bonds and fiscal difficulties by the government imposes losses on the central bank. At that time, suppose the central bank goes into bankruptcy. Does that imply inflation? Not really. It can pay off its existing liabilities with new ones--maybe pennies on the dollar. This is deflationary, but the reorganized central bank can then expand the quantity of base money again, presumably purchasing assets other than government bonds.
But what about the government's fiscal problems? Well, the central bank could inflate away the government's debts, but that is not necessary. Instead, the government could go bankrupt and pay off its creditors partially.
The reason to do this would be that inflationary default is still default on the government debt, but it also disrupts all of the other private contracts too. It is a massive externality.
Now the chance that central bank will explicitly default will reduce the demand for its monetary liabilities now. But that only means inflation now if the quantity of those liabilities is fixed. Otherwise, this possibility of explicit default in the future just means the demand for central bank monetary liabilities today is lower than otherwise. And it really isn't too much different from the possibility of inflationary default.
By the way, if the quantity of base money is taken as fixed, or on a constant growth path, then I will grant that worries about the ability of the government to keep up with its interest payments on government bonds would be inflationary. But notice that there is an implicit assumption of a money supply rule here.
As for the analysis of Krugman, he is equally wrong regarding the deflationary effect of budget surpluses. Of course, there, the problematic corner sultion is that if the demand for the monetary base rises more than the national debt, then open market purchases of government bonds won't be sufficient to maintain monetary equilibrium. However, as Beckworth notes at the end of his post, central banks can purchase other sorts of assets, such as foreign exchange. And, of course, there is always negative interest on reserves too.