Suppose your country (call it "Greece") is in recession, because there is an excess demand for money (call it "Euros"). And suppose that the Euro is both Medium of Account (prices are quoted in Euros) and Medium of Exchange (all other goods are bought and sold for Euros).
Now suppose your government introduces a new currency (call it "Drachmas"). It introduces the new currency by paying pensions via helicopter. But the Drachmas do not displace the Euros. The Euro remains as medium of account. Prices are sticky in terms of Euros, but the exchange rate between Drachmas and Euros is perfectly flexible, so prices are perfectly flexible in terms of Drachmas. Both Euros and Drachmas are used as media of exchange.
What happens? Does the introduction of the Drachma make the recession less severe?
I say it does make the recession less severe.
There are two ways to explain why it makes the recession less severe:
1. Unless the Drachma has an exchange rate of zero, the introduction of the Drachma, in addition to the existing stock of Euros, increases the total real value of the stock of media of exchange, and so lessens the excess demand for the media of exchange, and so reduce the severity of the recession. And if at least some people are willing to use at least some Drachmas as a medium of exchange, there will be some demand to hold Drachmas, and so the exchange rate of the Drachma will not be zero.
2. If at least some people are willing to use at least some Drachmas as a Medium of exchange, that means that Drachmas are an (imperfect) substitute for Euros. The introduction of a new good will reduce the demand for any existing good that is a substitute for the new good. So the introduction of the Drachma reduces the excess demand for the Euro, and so lessens the recession.
But in order to get this point, you have to understand the essentially monetary nature of recessions. They aren't caused by real interest rates being wrong, or real exchange rates being wrong, or real wages being wrong. Those are all just symptoms, or side effects. The underlying cause of the recession is the reduction in the volume of trade caused by an excess demand for the one good that is demanded or supplied whenever any other good is traded.
If you want a model, use my "Tiny money/macro model for microeconomists", and ask what would happen if we started out in recession, so there was too little trade in apples and carrots, and then introduce an endowment of a fourth good, dates, that was a substitute for the bananas that are used as a medium of exchange. Even if the prices of apples and carrots stayed exactly the same in terms of bananas, and the price of dates were perfectly flexible, the volume of trade in apples and carrots would increase.
This is in part a response to JP Koning's good post. He may be right about the Euro remaining the medium of account, but it is the excess demand for the media of exchange that causes recessions.