The circularity problem is that if the Fed trades based on the price of a futures contract, then the price of the futures contract will depend on expectations of the Fed's trades. That certainly looks like a circularity. The chairman of the Federal Reserve, Ben Bernanke, made this point years ago. (Bernanke & Michael Woodford, 1997. "Inflation Forecasts and Monetary Policy," NBER Working Papers 6157, National Bureau of Economic Research.)
As others have pointed out, this is an application of Goodhart's law. If the Fed starts targeting something, then the price begins to reflect what the market expects the Fed to do.
With index futures convertibility, where the money issuer (or issuers) buys and sells the contract, there is not a circularity problem exactly, but it is related. Under certain circumstances, people with relevant information don't trade the futures contract.
Suppose the Fed cannot change base money except to reverse a long or short position. Perhaps we can call this a passive or automatic version of index futures convertibility.
If everyone (perhaps including the Fed) knows that base money needs to increase some amount, say exactly $100 billion, or else NGDP will be exactly 1% below target, then no one will trade. If anyone did sell the future, then the Fed could, and must, buy $100 billion worth of bonds, raising base money $100 billion. Then those who sold the futures contract would buy (because now they expect NGDP to be on target) and the Fed would be hedged. No profits were made by the "speculator." Now that base money is exactly where it should be, no one has a position on the contract.
In the circularity argument, the price of the future actually would fall (supposedly) but the Fed would buy T-bills, expand base money, and get it back to target. But because the market expects this, the price would never actually fall. The price of the future would stay on target without the Fed buying or selling them, because everyone expects the Fed to do ordinary open market operations to get it back to target.
But if we don't treat "the market" as a representative agent, but rather as bulls and bears, then this problem doesn't arise. Or at least not always. Those who think that base money needs to increase by more than $100 billion sell the futures contract, and those who think it should rise by less than $100 billion buy the futures contract. The Fed is hedged when base money increases by $100 billion and NGDP is expected to be on target.
More importantly, from the perspective of traders, base money is always at the level where the market expects NGDP to be on target, and when a speculator expects something different, he or she trades. Even if everyone know this same thing, unless everyone knows that everyone knows, they still trade.
Of course, if keeping expected NGDP on target means that NGDP always will be on target, (so the system works perfectly) then no one will trade. If the Fed is on automatic, or passive, then the system would fail.
So, what a puzzle. Everyone knows that base money needs to increase by $100 billion, and if it increases by $100 billion, then NGDP will certainly be exactly on target. No one trades because if they do, there will be no profit. The Fed cannot increase base money unless "speculators" trade. There are no profits from trading. NGDP comes in below target, and this is known in advance, with certainty!
Now, if the Fed is allowed to adjust base money subject to the constraint that it remains hedged, these "everyone knows" scenarios, including the "with certainty" variant aren't a problem. The Fed increases base money by $100 billion. We might call these variants "constrained discretion" index futures convertibility.
The problem still exists if the Fed thinks that base money needs to increase $90 billion, and everyone else (the market) knows the needed increase is $100 billion, and further, everyone else knows that everyone else knows. Then they don't trade, because if they do, and the Fed is corrected, then there are no profits. So, the Fed is allowed to operate without the constraint binding because "the market" knows too much. Surely that is a bit odd. That modeling with representative agents makes this unusual situation into the norm is a failure of representative agent modeling.
I would also note that if we don't assume that all speculators are always self-interested, then in the everyone knows scenarios, it just takes one public spirited citizen to trade (bearing solely transactions costs) to solve the problem.
This would be important if the Fed knows exactly what it is doing, but chooses to keep NGDP away from target. Perhaps it expands base money in a way that will leave expected NGDP above target to help fund the national debt. Just one fiscal conservative who is willing to bear the transactions costs of a trade, and the Fed is forced to keep to the target.
Sumner has proposed trading contracts for very short periods of time, say a day, with value of NGDP for a day calculated as a weighted average of the past and future value. He has sometimes proposed keeping the trades secret, a bit like sealed bids (and offers, of course.) To some degree, these modification are efforts to avoid the circularity problem. Another reason for the modifications is to limit last minute trading by speculators so that they will avoid "showing their hand" to the Fed.
While further research on all of these problems is worthwhile, I think many of these problems involve unrealistic assumptions that "the market" is a single individual, and that the scenarios described in my post, where we imagine "Taylor" and "Krugman" taking different positions on the contract because they have quite different views, is the world in which we live.