It will be a 3 good model . It won't work with 2 goods, and 4 goods just complicates things and distracts attention from my point. 5 goods is right out.

It will be a one-period model . I don't need more than one period to explain my point; it just complicates things and distracts attention from my point.

It will be a model of a pure exchange economy . I don't need production in the model to explain my point; it just complicates things and distracts attention from my point.

I want a very simple model. The model will be extremely unrealistic. It won't look anything like the real world. But it will help me explain one very important fact about the real world.

I need the 3D Edgeworth box to be perfectly symmetric. That's important, and not just because asymmetry complicates things and distracts from my point. It's important because I want to introduce one asymmetry into the mechanism of exchange, and if the model is perfectly symmetric in all other respects it will be obvious that any asymmetry in the results must be caused by the one asymmetry I have introduced.

One of the 3 goods will be used as the medium of exchange. People can exchange A for B, and B for C, but cannot directly exchange A for C. That is the only asymmetry in the model, and I want to compare what happens with and without that asymmetry.

I need to tell a silly story to "motivate" that asymmetry. Let's see: only good B can be transported from one location to another, while goods A and C must be consumed at the locations where they drop from heaven. And people are anonymous, so I can't promise to give you some of my C if you give me some of your A. That silly story is good enough.

So there are two versions of the same model: a symmetric barter version, with 3 markets, where all 3 goods are transportable; and an asymmetric monetary version, with 2 markets, where only good B is transportable.

If Pa = Pb = Pc we get exactly the same market-clearing equilibrium in both versions of the model. The equilibrium is the point right in the centre of the 3D Edgeworth box. The monetary asymmetry doesn't matter.

What happens if prices get stuck at (say) Pa = Pc = 2Pb ? (The price of B is half what it should be, relative to goods A and C.)

For simplicity: assume the endowment point is at one of the corners of the Edgeworth box (each person has an endowment of 300 of one of the 3 goods, and zero of the others); assume everyone has preferences U=log(A)+log(B)+log(C) (so their Income Consumption Curves are Pa.A=Pb.B=Pc.C); and there's a thousand of each of the three types of agent (so we can talk about competitive markets).

Those with an endowment of 300 B's get to consume exactly what they want to consume at those non-market clearing prices, because Pb is too low, so there's an excess demand for B's, so they can sell as much B as they want. They choose to consume 50 A's, 100 B's, and 50 C's.

What about those with an endowment of 300 A's, and those with an endowment of 300 C's? Here's where the two versions of the model give different results.

In the barter version of the model, they each consume 125 A's, 100B's, and 125 C's. They want to buy more B's, but they can't find a willing seller of B's at the disequilibrium prices.

In the monetary exchange version of the model, it gets weird. Those with an endowment of 300 A's consume 200 A's, 100B's, and 50 C's. Those with an endowment of 300 C's consume 50 A's, 100B's, and 200 C's. They consume too much of their own endowment and too little of each other's endowment. The A-owners buy as much C as they want, but can't sell as much A as they want. The C-owners buy as much A as they want, but can't sell as much C as they want.

The B-owners get exactly the same utility in both versions of the model (but they get less utility than at the market-clearing equilibrium). The A-owners and C-owners get less utility in the monetary exchange version of the model than in the barter version of the model. And it's not because the relative price of A to C is wrong, because Pa/Pc=1, which is exactly right. It's because there's not enough trade in A and C, and that's because there's an excess demand for the medium of exchange B.

The A-owners would like to offer the C-owners a deal: "I will buy 75 more of your C's for 150 B's, if you agree to buy 75 more of my A's for 150 B's in return". And the C-owners would like to accept that offer. But they can't make that deal stick, because individuals are anonymous.

That's what real world recessions look like. Unemployed workers are stuck consuming their own endowment of labour, because they can't sell it for money. But anyone with money can buy as much as they want to buy. If the unemployed plumber and unemployed electrician could easily barter their labour, they would. But they can't, because barter is very very hard in a real world economy, with thousands of different types of labour.

It's got nothing to do with "too much saving". There is no saving in this model. It's a one-period model, dammit. Everything gets consumed during the period, then they die and time ends.

It's got nothing to do with the real interest rate being wrong. There is no interest rate in this model. It's a one-period model, dammit.

It's got nothing to do with relative prices of non-money goods being wrong, because they aren't wrong in my example. It's the price of money in terms of goods (the prices of goods in terms of money) that is wrong.

I think this model is simpler than "The World's Smallest Macroeconomic Model", because mine is a genuinely one-period model. Plus, TWSMM doesn't explain why unemployed workers can't do just as well working for themselves and consuming the goods they produce.

[Trivia: Roger Farmer may remember that day decades ago when we saw a real 3D metal box, with "Edgeworth" written on the lid. Ironically, the owner wouldn't sell.]

[This is a corrected revised version of what I was trying to do in an earlier post, where I messed it up. I think I got it right on this second attempt.]