After I wrote this I saw that Frances Coppola has a post that covers some of the same ground, but the point I want to make is different.
One of the things that made monetarism so popular until governments actually tried it was its simplicity. You can express this simplicity in many ways, but most involve the idea that there is a stable demand for the real value of money (M/p), so if you can control M you must control p. Never mind that the immediate influences on inflation were much more complicated: if you knew what M was, you would know what p would be. If you controlled M you would eventually control p.
There are lots of problems with this idea. I talked about the difficulty in explaining prices by just using money in this post. The difficulty of finding the ‘right’ definition of money is not a technical problem but a feature: because money can be saved as well as buy goods (the medium of exchange is also a store of value) focusing on its role in buying goods (‘hot potatoes’) is misleading. But even if there was a stable long run demand for money for some definition, the usefulness of this becomes questionable if we cannot say what the future quantity of money will be.
This becomes blindingly obvious if money is base money and we think about Quantitative Easing. Printing base money under quantitative easing does not imply hyperinflation because the expansion in the monetary base will be reversed once the recession is over. Knowing what base money is becomes useless as a tool for saying what future prices will be. (For those more technically minded who still think there is a Pigou effect, I discussed why the Pigou effect has disappeared from modern macro here. It is based on the same point.)
The Fiscal Theory of the Price Level is potentially another simplistic theory of inflation. This works from the identity that the real value of government debt must equal the discounted value of primary surpluses (taxes less government spending). It also can be used in a naive way: treat future primary surpluses as fixed, and any increase in nominal government debt must lead to higher prices. But, as Chris Sims explains in this nice exposition at Lindau, future primary surpluses are not fixed. If debt increases, future primary surpluses can increase to pay the interest on that additional debt, and more.
There may be some that say that we cannot trust politicians to do that. To which I say which planet have you been on for the last five years? As Brad DeLong reminds us for the US, this recession has been unusual in the zeal that governments have shown in rapidly reducing primary deficits, and of course in the Eurozone this zeal - embodied in the fiscal compact - has led to a second recession. Chris Sims raised the possibility that so great has this zeal been that even though nominal debt has risen, the price level might fall to make the identity hold.
One lesson I would draw from this is that the Fiscal Theory of the Price Level, like monetarism, is not a terribly helpful way of thinking about future inflation. The idea that we can take one variable, or one equation, and distil from that the future price level is a fantasy. What is surprising is that this fantasy has been, and still remains, so attractive for some economists.