There are two parallel worlds. Both worlds use bits of coloured paper as money, because barter is very difficult. The green world uses green paper as money, and the red world uses red paper as money. The green paper money has a positive value. The red paper money has a negative value.
In the green world, if I sell you apples, you agree to give me your green money in exchange. Goods and money flow in opposite directions around the economy. In the red world, if I sell you apples, you agree to take my red money in exchange. Goods and money flow in the same direction around the economy.
In the green world, it is illegal for anyone except the central bank to create money. And only the government is allowed to take your money without your consent.
In the red world, it is illegal for anyone except the central bank to destroy money. And only the government is allowed to give you money without your consent.
These laws are much harder to enforce in the red world than in the green world.
In the green world, people will sometimes lend money, in return for a promise to give back that money, plus interest, at some future date.
In the red world, people will sometimes borrow money, in return for a promise to take back that money, plus interest, at some future date.
In both worlds, people are not allowed to have negative net wealth, in case they die leaving a negative bequest to their heirs.
This law is much harder to enforce in the red world than in the green world.
In both worlds, the central bank, which issued the paper money, may decide to pay positive (or negative) interest to people who hold that paper money, by giving them more money (or taking money away).
Negative interest rates are hard to enforce in the green world, and positive interest rates are hard to enforce in the red world.
MV = PY works fine in both worlds, except both M and P are negative in the red world.
But slightly different things happen if M halves, so there is a shortage of money at existing prices.
If there is a shortage of money in the green world, there is a fall in trade, and an excess supply of goods, because people stop buying goods. This causes sellers of goods to cut their prices (prices halve), so less money is needed to buy the same volume of goods, so the shortage of money is eliminated.
If there is a shortage of money in the red world, there is a fall in trade, and an excess demand for goods, because people stop selling goods. This causes sellers of goods to raise their negative prices (prices halve), so less money is needed to sell the same volume of goods, so the shortage of money is eliminated.
Then, in both worlds, the central bank decides that it is too much of a hassle enforcing all the laws. So it insists that all money be deposited in the central bank to make it easier for the central bank to monitor who owns what money, to prevent unauthorised creation, destruction, and transfer of money, and to make it easier to pay interest. The central bank creates a box for each person, puts their money in that box, and transfers money between those boxes when it gets their consent to make those transfers.
In both worlds, a rumour spreads that the central bank has actually destroyed all the money, and is simply keeping a record on a ledger about how much money is supposed to be in each person's box. But economists say that rumour has no empirical content; it doesn't matter if it is true or false.
Then the two parallel worlds merged into one world. The combined world used both green notes and red notes as money.
If the combined central bank finds a person has 6 green notes and 2 red notes in his box, it tears up 2 green and 2 red notes, leaving 4 green notes. If the combined central bank finds a person has 2 green notes and 6 red notes in his box, it tears up 2 green and 2 red notes, leaving 4 red notes. And when it adds up all the notes across all the people, it finds the total number of green notes is exactly the same as the total number of red notes.
And then some New Keynesian macroeconomists said that money didn't exist in the new combined world. They said there was no "cash", only "credit".
They were wrong.
[I think I've got this right. But the bit about halving the money supply was hard to get my head around.]
Worthwhile Canadian Initiative 
Frank Restly: “In the red world, why wouldn’t both receipts go to the purchaser of a the car?”
The thing speaks for itself.
Frank Restly: “Purchaser of car receives car, red notes, receipt for car received, and receipt for red notes received. And you are assuming that the red world operates such that a receipt is recognized by the courts as evidence. Having no means to prevent forgery of receipts (central bank / government prevents forgery of red notes), the courts in the red world could rule them invalid.”
Forgery is a problem in any world. The Chinese used to use fingerprints (thumbprints, IIRC) instead of signatures, or in addition to them. The personal stamp, which is still in use, is also not easy to counterfeit.
If green notes are backed by government debt, then I suppose red notes are backed by government taxes. The government presents everyone with their tax bill in red notes and people sell their goods to pass off their bill to those selling to government.
Min,
“In the red world, why wouldn’t both receipts go to the purchaser of a the car? – The thing speaks for itself.”
I agree that giving both receipts to the purchaser of the car would be strange / unusual, but it would be no stranger than negative money.
“Forgery is a problem in any world.”
Agreed. And so a legal system has to deal with the possibility of forgery. It can rely on government sponsored anti-forgery devices (fingerprint identifiers, personal stamps, etc.) or it can preclude items that are subject to forgery as evidence.
One other thing – suppose in this red world, the government / legal system has an anti-forgery receipt detector – for instance all receipts must be printed on special green paper with special insignias all provided by the government. Are receipts green money in a red money world?
@Nick:
Majromax: If you originally had 10 red, you could mow the lawn for 10 red, then go buy a meal for 10 red. You end up with the same 10 red as you started, but a full belly in exchange for mowing the lawn. If you miss an opportunity to mow a lawn, you miss an opportunity to eat a meal, if you keep your beginning and ending stock of red the same.
@Nick: