How can the Cheshire Cat disappear, but its smile remain?
How can money disappear from a New Keynesian model, but the Central Bank still set a nominal rate of interest and create a recession by setting it too high?
Ignore what New Keynesians say about their own New Keynesian models and listen to me instead. I will tell you how it is possible.
Start with a model of an economy with a central bank that issues paper currency. It's a monetary exchange economy, where that currency must be used to buy and sell everything else. Barter is assumed to be prohibitively costly. The currency issued by the central bank is the sole medium of exchange. It is also the unit of account. The central bank can change the stock of currency in circulation by using open market operations.
Now replace the paper currency with chequing accounts at the central bank. Instead of having $100 recorded on paper notes in your pocket, you have $100 recorded on silicon in your account. Instead of transferring paper notes from pocket to pocket when you buy something, you pay by cheque or debit card and the central bank transfers the silicon money from account to account. It makes no difference to the model.
Now suppose the central bank pays interest on the balance in your chequing account at the central bank. There is no puzzle about how the central bank can set that interest rate; it just does it. (The central bank could have paid interest on the currency it issued, but that would have been administratively more difficult.)
Now suppose the central bank allows people to have a negative as well as a positive balance in the chequing accounts. You can run an overdraft. And suppose the interest rate you pay on a negative balance is the same as the interest rate you receive on a positive balance. So the central bank only sets one interest rate.
Now suppose the central bank uses open market operations to adjust the net stock of money until the sum of the positive balances exactly equals the sum of the negative balances. The central bank now can have zero net worth, because the negative balances are its assets and the positive balances its liabilities and both are equal. The central bank can now have zero net income, because it pays out interest on positive balances equal to what it earns on negative balances.
Now suppose the economist building the model economy decides to make all agents identical, so that all agents always hold identical stocks of money balances. If one agent spends $100, all other agents do exactly the same at exactly the same time, so $100 is both added and subtracted from his chequing account at exactly the same time. But in symmetric Nash equilibrium, each agent nevertheless chooses how much money to spend taking as given others' choices of how much money to spend. And each agent's choice of how much money to spend depends on the rate of interest paid on holding money that is set by the central bank.
The above two paragraphs together mean that each agent always has a $0 balance in his chequing account at the central bank. But the rate of interest set by the central bank still affects spending.
Start in full equilibrium, which is where the economy would be if all prices were flexible. Now, just to keep it simple, hold all prices fixed at that level. If the central bank sets the interest rate too high, people spend less money, and so each sells less goods to other people. There's a recession. By assumption, all relative prices are correct, and there are unexploited gains from trade at those relative prices. If people could barter their way back to full employment they would immediately do so, to exploit those gains from trade, and the recession would end as soon as it began. But the real interest rate paid on holding money is too high. If two agents could cut a deal, whereby each agrees to buy $100 of the other's goods if and only if the other buys $100 of his goods, so the deal leaves both with unchanged money balances, they would do so. But that deal is equivalent to barter, and is ruled out by assumption.
The Cheshire Cat has disappeared, but its smile remains. And its smile (or frown) has real effects.
The New Keynesian model is a model of a monetary exchange economy, not a barter economy. The rate of interest is the rate of interest paid on central bank money, not on bonds. Raising the interest rate paid on money creates an excess demand for money which creates a recession. Or it makes no sense at all.
[I've said all this before, but I've said it simpler and better this time.]