I'm writing this post mostly to try to get my own head straight. Read at your own risk.
Strategy space matters. The order of moves matters. What I call "the Simple Money Game" is a three-stage mixed Bertrand-Cournot general equilibrium game. Bertrand moves are made before Cournot moves. Each player is both a producer of one good and a consumer of many goods. The producer of one good plays Cournot, and the producers of the other goods play Bertrand. Consumers always play Cournot. That one good serves both as medium of account and as medium of exchange. "Money" is defined by the strategy space. Or maybe vice versa.
I think this Simple Money Game is similar to what many macroeconomists have at the back of our minds. It's similar to a New Keynesian model in particular. But I want to use it to look at the difference between money and other goods, and between the producer of money and the producers of other goods.
In Bertrand-Nash equilibrium, each producer sets the price of his good.
In Cournot-Nash equilibrium, each producer sets the quantity of his good.
We can also imagine a mixed Bertrand-Cournot-Nash equilibrium, where some producers set price (play Bertrand) and other producers set quantity (play Cournot).
Normally we talk about these games in a partial equilibrium context. But I want to talk about a general equilibrium context.
We cannot have a general equilibrium version of the Bertrand-Nash game. If there are n goods, there are only n-1 prices, so we cannot have n producers each setting his price. If there are only two goods, the price of apples in terms of bananas is by definition the reciprocal of the price of bananas in terms of apples. We cannot have the apple producer and the banana producer both setting the relative price of apples and bananas. If they try to set different prices, they simply create an opportunity for infinite arbitrage profits. One will give up, and play Cournot instead.
The "Simple Money Game" is a general equilibrium game with n players and n goods, with each player producing one good and consuming many goods. The producers of n-1 of those goods play Bertrand. Each sets a price of his good in terms of the nth good, and will accept payment only in the nth good. This means the nth good is both Medium of Account and Medium of Exchange. The nth good is money. The producer of the nth good plays Cournot. He decides what quantity of the nth good to sell.
We could say that each of the n players plays Cournot when they decide how much to buy with money, because they are choosing a quantity. But the n-1 producers of the non-money goods play Bertrand when they choose a price to sell their goods for money. In the normal ways we use the words (we never talk of "buying or selling money", forex markets aside) the n-1 players are both buyers and sellers. The nth player is only a buyer. Sellers play Bertrand; buyers play Cournot.The order of moves matters. No player will announce how much he will buy before the seller announces the price, because this would give the seller an incentive to announce an infinite price.
One possible assumption is a two-stage game: all players announce their prices simultaneously in the first stage; and then announce their quantities simultaneously in the second stage. In this game, if the producer of money announces that he will sell less money than the other players expected, some of the other players may be unable to buy the quantities of goods they had announced they would buy. They simply run out of money (hit the non-negativity constraint), and cannot buy what they said they would buy. And this effect will snowball.
But instead I want to assume a three-stage game. All players announce their prices; next the producer of money announces his quantities; finally the other players announce their quantities. (You could say this is a variant on the Stackelberg game, with n-1 players being both leaders and followers, and the nth player coming in between.) [Update: I am implicitly assuming production takes place after buyers have placed their orders, and you might want to call this a fourth stage of the game.]
In this three-stage version of the Simple Money Game, we get a recession if the producer of money announces a smaller quantity of money than the other players had expected. Each agent knows he will sell fewer goods to the producer of money than he expected, so will revise down his own planned purchases of goods because of this, and he knows that other players will do the same, which means he will expect to sell even fewer goods, so he revises his planned purchases down because of this, and so on. The Nash equilibrium will have a smaller quantity of goods sold.
If this process happened in real time, it would sound similar to an Old Keynesian multiplier process, or an Old Monetarist hot potato process. But it all happens in virtual time in a one-shot Nash equilibrium game.
I think this three-stage version of the Simple Money Game is roughly what most of us macroeconomists have at the back of our minds.
We could have a repeated version of the Simple Money Game, where some or all of the goods are durable, with depreciation rates less than 100% per period. We would want to assume the nth good is durable.
Each producer of the n-1 non-money goods has n-1 market decisions each period. He decides the price of his own good. And he decides how much to buy of each of the other n-2 non-money goods. The producer of the nth good also has n-1 market decisions. He decides how much to buy of each of the n-1 non-money goods. (And every producer also has one non-market decision: he decides how much of his own good to produce for his own consumption).
In the repeated game, each producer of the non-money goods also has a long run money constraint. If he spends more money buying the n-2 other goods than he earns selling his own good, his stock of money will fall over time, and eventually hit the non-negativity constraint. In the long run he only has n-2 independent market decisions.
The producer of the money good has no such long run money constraint linking his market decisions. Each period he simply decides how much of the n-1 non-money goods to buy.
If we added uncertainty in the repeated game, the n-1 players would have an incentive to hold money even if holding it gave them no direct utility. It would reduce the risk they would hit the non-negativity constraint if they had unexpectedly few buyers for their good in one period.
All players also have a "production constraint" linking their market decisions, and their decision on how much of their own good to consume, to their total production. Each must produce enough both to sell and for his own consumption.
In Walrasian General Equilibrium theory, each agent has a budget constraint that is exactly like the production constraint of the producer of money in the Simple Money Game. He decides how much to buy of all the other goods, taking prices as given. He must produce enough of his own good to sell in exchange for what he buys, plus his own consumption. As others have said before: Walrasian General Equilibrium theory treats all goods as if they were money. But as I have shown, the n-1 players in the Simple Money Game who are not producers of money have a long run money constraint linking their market decisions in addition to their production constraint.
We could change the assumptions to allow each player to produce two goods. The second good would be an IOU, signed by the player, to deliver the nth good at some future date.
We could also change the assumptions so that prices could not always be changed in every period. That would allow recessions to last for many periods.
If we made both those changes, the Simple Money Game would be something like a New Keynesian model.