How does monetary policy work? Think of it this way: Fiscal policy adds or subtracts government liabilities. Monetary policy swaps government liabilities.
You run a bigger deficit and you have to borrow more to pay for it. You issue more Treasury securities.
The central bank can exchange some of its liabilities for Treasuries. Treasury securities bear interest. Traditionally, central bank liabilities do not. Think of cash, for example.
People will accept cash in exchange interest-bearing debt, but they would rather hold interest-bearing debt than cash. That is, interest-bearing debt is a better long-term store of value--they only want cash as a short-term store of value, i.e. as a medium of exchange. Therefore, it seems reasonable to assume that the velocity of cash will be higher than the velocity of interest-bearing debt, so that you get more spending when the central bank swaps cash for securities. I am not really so sure about this, but I think it is what the traditional story of a monetary expansion amounts to.
Frequent commenter Winterspeak seems to want to say that the Fed can eliminate the debt at the stroke of a pen. Well, it can trade non-interest-bearing liabilities for interest-bearing liabilities. My guess, though, is that people do not want to store their wealth in non-interest-bearing securities, so that if the Fed did this a lot we would see a lot of spending and a lot of inflation, and that inflation would transfer wealth away from current holders of U.S. debt and toward future taxpayers. It would destroy a lot of people's life savings while getting rid of a lot of future taxpayer liabilities.
The traditional MV = PY story is that in the long run the price level is proportional to the amount of non-interest-bearing debt in circulation. I am not as comfortable with this story as the typical economist. I think that prices are determined by habits and market conditions.
Suppose (and I am not sure I like this, either) we change the traditional story to have M be a weighted average of all outstanding government liabilities. The traditional theory puts a weight of 1 on non-interest-bearing debt and a weight of 0 on interest-bearing debt. Maybe the more general case is to put a weight of less than 1 on non-interest-bearing debt and a non-zero weight on interest-bearing debt. This treats the various forms of debt as less distinct than the traditional theory implies.
Note that last year the Fed started to pay interest on reserves, and that the main instrument of monetary policy is to swap reserves for Treasuries (it's actually is a more subtle transaction than that, using repos). So now we are exchanging two forms of interest-bearing debt, which might be an argument for the weighted-average approach.
From this perspective, the Fed is doing debt management. The Treasury does debt management when it decides on the mix between long-term and short-term debt. The Fed does debt management when it decides to swap its liabilities for Treasuries.
I think that looking at what the Fed does as debt management might be insightful. I think it helps to play down the significance of monetary policy, which I am constantly trying to do. Whether the weighted-average notion of M is an improvement is an issue I still need to noodle over.
I will grant that once inflation gets high enough (5 percent? 10 percent?), the distinction between interest-bearing debt and non-interest-bearing debt becomes increasingly important. People become less prone to hold wealth in non-interest-bearing assets and therefore more prone to spend non-interest-bearing cash more quickly. Thus, weighted-average M begins to look at lot like regular old M when the inflation rate is sufficiently high.