The mechanics of exchange and non market-clearing prices

[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.)

Comments

[Majromax]

I don't think your model is complete here. It barely works in equilibrium, but it requires multiple stages of trade that you neglect from the sticky-price case.

We have a fruit barter economy where all trades must go through the banana producer (and involve bananas). There is also a fruit-in-advance constraint, so that fruit cannot be traded on credit. In equilibrium, each agent begins with 3 tons of its own fruit and wants to end up with 1 ton of each. This is achievable via:

* The apple producer swaps 2 tons of apples for 2 tons of bananas. (Stocks: (1,2,0), (2,1,0), (0,0,3))
* The carrot producer swaps 1 ton of carrots for 1 ton of bananas. (Stocks: (1,2,0), (2,0,1), (0,1,2))
* The apple producer swaps 1 ton of bananas for 1 ton of carrots. (Stocks: (1,1,1), (2,1,0), (0,1,2))
* The carrot producer swaps 1 ton of bananas for 1 ton of apples. (Stocks: (1,1,1), (1,2,0), (1,0,2))
* The carrot producer swaps 1 ton of carrots for 1 ton of bananas. (Stocks: (1,1,1), (1,1,1), (1,1,1))

I do not think that this convoluted sequence of trades is precisely what you had in mind with your hypothetical, but I think it is necessary to preserve the "bananas are money" allegory.

This sequence of trades also requires the banana producer to accept trade that will temporarily decrease its utility. If the banana producer would only accept trades to the extent that they were utility-inproving (Nash equilibrium versus Pareto optimal), then it would only trade a gross 2 tons of bananas, and our Nash-equilibrium stock distribution would look like ((2,1,0), (1,1,1), (0,1,2)).

If the banana producer can act with foresight, however, then I don't think the situation is as futile as you suggest with disequilibrium sticky prices.

[Nick Rowe]

Majro: good man! My brain is too old and tired.

[Nick Edmonds]

Nice idea, but I think I'd agree with Majromax that it's not quite there.

How about some slight modifications: Each agent also produces 3 tons of dates. Each agent only consumes dates and one other fruit (not being his own produce, with a no-double-coincidence triangle).

The optimal solution involves all apples, bananas and carrots being traded. If open three way exchange can take place, this only requires that the relative price of those three fruit is correct. The price of dates is irrelevant.

However, if all exchange involves dates (say agents meet individually and know prices, but do not know about demand for fruit that they don't themselves consume), the relative price of dates now matters.

[Mike sproul]

Agreement with Nick Rowe. What a strange sensation!

Oh sure, I would have said that transaction costs are like a tax on trade, and money lowers that tax and avoids a deadweight loss, and I would have had supply curves sloping up instead of vertical, and I wouldn't talk about sticky prices, and...oh wait, maybe there's still some disagreement after all.

[Jussi]

The government can tax (confiscate if you will) bananas and redistribute the fruits in a pareto optimal way to escape the local optima suggested by Nash.

[Nick Rowe]

I probably posted this a bit prematurely. Should have saved it first, then thought about it. Oh well, my old brain is fried from having to get the grades in, using three different sorts of spreadsheet-like programs. I will let you younger and smarter people try to figure it out.

[Frank Restly]

Majromax,

"If the banana producer would only accept trades to the extent that they were utility-inproving (Nash equilibrium versus Pareto optimal), then it would only trade a gross 2 tons of bananas, and our Nash-equilibrium stock distribution would look like ((2,1,0), (1,1,1), (0,1,2))."

You can still get the desired distribution.
Apple producer sells 1 ton apples for 1 ton bananas
(1, 2, 0) (1, 1, 1) (1, 0, 2)
Carrot producer sells 1 ton carrots for 1 ton bananas
(1, 1, 1) (1, 1, 1) (1, 1, 1)

[Jussi]

@Nick: "Should have saved it first, then thought about it."

I hope this is not meaning less "I have not yet got my head around" posts, which have been great in many ways. There is a risk of erring every now and almost never but even here value was added. Please keep the ideas and posts flowing.

[Jussi]

@Frank:

Apple producer sells 1 ton apples for 1 ton bananas
---> (2, 1, 0) (1, 1, 0) (0, 0, 3)
Carrot producer sells 1 ton carrots for 1 ton bananas
---> (2, 1, 0) (1, 1, 1) (0, 1, 2)

brackets for:

(Apple prod) (banana prod) (carrot prod)

and inside of brackets:

(Apple quantity, banana quantity, carrot quantity)

[Nick Rowe]

In order to have an incentive to act as middlemen (both buying and selling apples), banana producers need to be able to buy apples from the apple producers at a slightly lower price than they sell apples to the carrot producers. Similarly, they must be able to buy carrots from the carrot producers at a slightly lower price than they sell carrots to the apple producers.

If all three types are arranged in a triangle, you only need one price for each good. If they are arranged in a row, you need two prices for each good.

[Nick Rowe]

Jussi: thanks. But this was one of my least thought-out posts.

[Kevin Donoghue]

Nick,
One who has an endowment of 3 tons of apples should not be called an apple producer. The distinction between a monetary exchange economy and a barter economy clearly matters a lot to you. To me, the distinction between a pure exchange economy and an economy where people actually work seems much more important. In the latter, gloomy expectations will give rise to a depression, whether money is used or not. The autumn's apple harvest depends on spring forecasts of banana and carrot production.

[Nick Rowe]

Nick E: I think your model works, but it's not symmetric. What I was looking for was a model where there was a perfect symmetry between all goods, so that with barter or Walrasian auctioneer they are interchangeable. But where a trivial difference made one of those goods the medium of exchange, so we could isolate the effects of being a medium of exchange.

[Nick Rowe]

Kevin: we don't have a name for "a person who has an endowment of apples".

More substantively, what is the difference between:

1. A person who owns an apple tree that produces 3 tons of apples per year regardless, which if he cannot sell he can only consume himself.

2. A person who owns his own body, that produces 24 hours of labour per day regardless, which if he cannot sell he can only consume himself.

The hairdresser, manicurist, and masseuse are like 3 people who each own one tree that produces one of 3 different types of fruit.

Unemployment does not mean less labour; it means less labour is sold, so the unemployed have to work for themselves. Recessions are a drop in the volume of trade, and that's what makes people worse off. It makes no difference whether labour (and land) produces physical goods or non-physical services.

[Kevin Donoghue]

Nick,
In Ireland, Tony O'Reilly is popularly known as the Bean Baron, so I think we may speak of apple moguls and banana sheikhs in your imaginary world. If we're thinking about macro there is an important difference between those who sit on a resource which they can sell or not, as they please, and those who must plan a production process. Notice that in the FX market the mechanics of exchange frequently involve using a major currency to create a sale of, say, New Zealand dollars for Kuwaiti dinars. But this seldom causes the kind of problems we see in a slump. You say unemployment does not mean less labour. That may be a defensible simplification in a model, but in reality mass unemployment typically does not result in a massive surge of DIY activity.

However this may all be a bit beside the point. I think the crucial thing is that you want a model to be clearly labelled "monetary exchange economy" or "barter economy" - that doesn't strike me as a reasonable demand. A model may not fit comfortably into either category, without being a bad model. I think Krugman lays out the choices more clearly: "models with rational expectations, markets continuously in equilibrium, and unique equilibria don’t cut it. But which pieces of such models would you want to modify or replace?" You clearly think that the appropriate modification involves money's role as a medium of exchange, but I can't see why Jordi Gali, for example, has to agree with you.

Krugman did another post sort-of related to all this, with an interesting reminiscence:

Incidentally, let me chime in with an example of an issue Simon, Brad DeLong, and others have also raised: the peculiar selectivity of what is considered to be true microfoundations. When I was an assistant professor at Yale, I distinctly remember hearing Martin Shubik declare that all the great early analyses of general equilibrium, including Hicks and Arrow, were illegitimate, because they simply assumed a price mechanism. They should, he insisted — or at least that’s how I remember it — have waited until theorists proved that competitive equilibrium is the core of a noncooperative game with many players.

If you say that this is silly — and I agree — you then have to ask why. What determines which things must be derived from first principles, and which can simply be assumed?

I vaguely recall that, around the same time, Karl Brunner was telling Arjo Klamer that there are no first principles. I tend to agree.

[Nick Rowe]

Kevin: If someone builds a macro model simply saying "assume that all other goods (including labour and bonds etc.) can only be bought or sold for money" I would be a lot happier. I want that assumption (or some alternative assumption) to be explicit. Because it matters. If it can explain the reasons why barter is very costly, that's icing on the cake, but not essential for every macro model. Understanding why people use monetary exchange is an important question, but not every model has to answer that question.

"You say unemployment does not mean less labour. That may be a defensible simplification in a model, but in reality mass unemployment typically does not result in a massive surge of DIY activity."

Mass unemployment results in the unemployed either doing DIY production or else consuming leisure. Just like the apple producer who cannot sell his apples must either bake DIY apple pies or else consume the apples himself.

"Leisure" is just labour we consume ourselves.