[Update: on second thoughts, maybe this post was not quite ready for prime-time. But I think it's still fun to play with.]
Here is a very simple model of a pure exchange economy where the mechanics of exchange (who can trade what with whom) determine whether very small departures of prices from market-clearing cause very large welfare losses.
There are three goods, apples, bananas, and carrots, with prices Pa, Pb, and Pc.
There are three types of agents: 100 apple producers with an endowment of 3 tons of apples each; 100 banana producers with an endowment of 3 tons of bananas each; 100 carrot producers with an endowment of 3 tons of carrots each.
All agents have the same preferences U = log(A) + log(B) + log(C).
Agents of type t maximise utility where At.Pa=Bt.Pb=Ct.Pc (they consume equal values of all three goods).
The resource constraints are: Aa+Ab+Ac=3, Ba+Bb+Bc=3, Ca+Cb+Cc=3
Competitive equilibrium is where Pa=Pb=Pc and Aa=Ab=Ac=Ba=Bb=Bc=Ca=Cb=Cc=1
(All three goods have the same price and each agent consumes one ton of each of the three goods.)
All obvious boring micro stuff.
Now for the mechanics of exchange.
1. Suppose that all three types are lined up in a row, abc, and that you can only trade with an agent who is next to you. So banana producers (in the middle) can trade with everyone, but apple producers cannot trade with carrot producers.
Now suppose that the price of bananas is stuck an epsilon below the market-clearing level: Pb=Pa-e=Pc-e, where epsilon (e) is a very small number.
Banana producers maximise their utility by: swapping slightly less than one ton of bananas for slightly less than one ton of apples, swapping slightly less than one ton of bananas for slightly less than one ton of carrots, consuming slightly more than one ton of bananas, slightly less than one ton of apples, and slightly less than one ton of carrots.
Banana producers get to consume exactly that bundle of goods they want to consume, taking the price vector as given.
Apple producers and carrot producers would like to consume slightly more bananas, but the banana producers won't sell them any more.
Apple producers consume zero carrots. Carrot producers consume zero apples. Because the banana producers (in the middle) don't want to do any more trades at the prevailing price vector.
2. Now suppose that all three types of agents are rearranged into a triangle, so that each type can now trade with both other types. But the price of bananas is (as before) still stuck an epsilon too low. The apple producers and carrot producers would immediately swap slightly more than one ton of apples for slightly more than one ton of carrots. Both get a large jump in utility relative to when they were lined up in a row abc.
The moral of this story is that bananas are money, in the sense that apples and carrots are traded for bananas, but apples are not traded for carrots directly. "Money buys goods, and goods buy money, but goods do not buy goods" (Clower). An excess demand for money can matter a lot. Even a very small amount of price stickiness can have large welfare consequences, if it causes the price of money to be too low, because it disrupts trade in all other goods. If the price of bananas were allowed to rise, relative to apples and carrots, banana producers would earn profits from acting as middlemen. Bananas need to be an epsilon overpriced, so there is an epsilon excess supply of bananas, for trade to go smoothly.
This is a model of recessions, in case you missed the point. The underemployed apple producer/worker has to consume too much of his own apples/labour, and consumes too few carrots/some other underemployed work's labour.
It's also a highly non-linear model, if that's what gets you off. It's a Milton Friedman plucking model, where recessions are bigger than booms.
It shows that monetary exchange matters. So stop building sticky price macro models where nobody can figure out if they are models of a monetary exchange economy or a barter economy.
(The precise math is left as an exercise for the reader. "Slightly less" and "slightly more" are good enough for me.)