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Excellent, Nick, and I'd agree this is much more on point than the earlier version.

I would say that your conclusion that saving is irrelevant is partly due to the fact that in this model the medium of exchange is a commodity that is valued purely for its current consumption value (apart from its exchange benefits). In modern economies the medium of exchange has no current consumption value; its value comes from its use in securing consumption at some later point in time, so the saving aspect is relevant.

Nick E: Thanks! It took me so much thought to finally get it right, even though (or maybe because?) it's such a simple model.

Suppose all 3 goods were (infinitely) durable goods. We got a flow of pleasure from owning them. An increased desire to save wouldn't affect anything. The way that saving usually matters in monetary models is we assume that the money good is durable and the goods that get unemployed are not. Which complicates things, so you can't see what is caused by the money/non-money difference vs what is caused by the durable/non-durable difference.

I think that's right.

Just as a possible extension, I think you could add a fourth commodity M into this so that U = log(A) + log(B+M) + log(C) and Pb=Pm. The aggregate quantity of M is zero but each individual can accumulate a positive or negative amount (s.t. M=>-B). Lastly, our exchange rule is that all trades must involve M as one leg, rather than B.

I think you should get the same outcomes, even though the excess demand is for B+M and there is no real shortage of M specifically.

Point being that what matters is not the medium of exchange alone, but the medium of exchange plus close substitutes.

Nick E: No! No extensions allowed! Because that would complicate the model!

But what I could do is add a shift parameter "m" to the utility function, so it reads U=log(A)+log(m+B)+log(C). Which would be the same as your extension. But it would complicate the model.

Nick, I think I need to read this a couple more times to really get it, but thank you - this is monetary theory I can understand.

I'm probably going to run afoul of some assumptions, but I am going to try anyway.

Let's call endowment the supply. Supply of A = 300, supply of B (MOE) = 300, and supply of C = 300. Now let's have real aggregate demand of A and C be 210 (70 each). Now run the model.

I'm pretty sure Nick is assuming real AD of A is equal to or greater than 300 and real AD of C is equal to or greater than 300. That sets up what I believe is called an aggregate demand shock.

"Suppose all 3 goods were (infinitely) durable goods. We got a flow of pleasure from owning them."

Is there any consumption then?

Hi Nick,

The three good Edgeworth box isn't actually 3D, but rather 6D.

The two good Edgeworth box is nominally 4D: 2 for each good and 2 for each person's allocation for a total of 2 x 2 = 4. There is a simplification where your endowment of A determines my endowment of A and likewise for B. Those two constraints bring the dimension down to 2D. And the form of those constraints allow us to identify, if A = A' + A'', that A'' = A - A' so that the x-axis is only in terms of A' and the global constant A and the y-axis is analogously only in terms of B' and the global constant B.

In the three good version, we have 3 goods and 3 people's allocation for a total of 3 x 3 = 9 dimensions. However, there are only 3 constraints (total of each person's endowment sums to the total of each good A' + A'' + A''' = A) leaving us with 6D. And in the three good case, the axes don't combine: A'' = A - A' - A''' so the first person's allocation doesn't tell us what the second person's allocation is.

So I don't think a three-good Edgeworth box is a simple generalization of a two-good one.

I wrote something on this last year which, although less formal, shows something similar I think: http://monetaryreflections.blogspot.co.uk/2014/05/the-problem-with-monetary-exchange.html

Nick, as Jason says you're into higher dimensions than you think. AFAICT there's no problem if you restrict yourself to just 2 representative agents and 3 goods. Then if we choose units such that the aggregate endowment of each good is 1 unit, there's a 1-dimensional contract curve going from (0,0,0), which is the "origin" from Agent A's viewpoint, up to (1,1,1) which is Agent B's origin.

We don't seem to have progressed very far with this discussion in 4 years:

http://anyoldbullshit.blogspot.ie/2011/02/nick-rowes-critique-in-pictures.html

Frances: Thanks! Trying to do my best for micro-macro relations!

TMF: "I'm pretty sure Nick is assuming real AD of A is equal to or greater than 300 and real AD of C is equal to or greater than 300."

Talk about missing the point. I'm not *assuming* demands for the 3 goods. I am *DERIVING* demands for the 3 goods, given prices. Why do you think I wrote down the utility function!

Jason: Yep. The 3D Edgeworth box really only works for 3 goods and 2 agents. It's squeezing in the third agent that is tricky. (Not that this affects the math representation at all.)

Nick E: I like your post. Your model makes the same point as mine. I think I have seen you do other posts that make the same point, in different ways.

The advantage of your setup is that you only need 2 types of agents. Each has an endowment of the medium of exchange. (So in my setup, I would delete the B-owners, and give the A-owners and C-owners an endowment of 150 B's each.) The only disadvantage is you lose the symmetry between the 3 goods. Not sure if that disadvantage is big enough.

Kevin: in your model, what are "employers" endowment and preferences? Is it the same model as I have sketched in the paragraph above?

Nick, I'm not pushing a model of my own, I'm just trying to see yours in 3D. I need to do some algebra to get a better idea. But not today. The summer seems to have shown up at last & I'm not betting on it sticking around for long.

A model in equilibrium would have specified prices. The change from Pa = Pc = Pb to Pa = Pc = 2Pb is a simple exchange of one equilibrium for another equilibrium.

In your text, you have three owners who receive 300 A or 300 B or 300 C. If allowed to pick between the three under conditions Pa = Pc = Pb, the choice would not matter. Under conditions Pa = Pc = 2Pb, the choice would matter because B the least valuable of the three groups of 300 items.

Now if we want to allow trade between the three groups, are we going to seek a result of equal value between the three participants ((Pa + Pc + 2Pb)/3), or are we going to seek equal consumption (Pa = Pc = Pb)?

In your example, B was the trade decider. B made a choice of 50 A, 100 B, and 50 C.

In your example, B (with the least value of endowment), made the choice of how much trade should occur. Notice that B also retained 1/3 of his endowment.

This model looks good, and I agree with the basic premise (having a medium of exchange can distort consumption). I do have an issue though - hopefully someone can clear it up:

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.)

I read this as "what if money halves in value", right? That's a monetary shock causing a recession, not an AD-driven one. So, for example, if money halves in value (because of expectations of future inflation or a massive devaluation) you will get a recession and the A/C market will fail to clear. If the value of the medium of exchange crashes, you can have knock-on effects in distribution in the real economy. Though this model is one-period, I think it could be used to explain short term effects.

When I think about Keynesian (AD) depressions though, money goes up in value, not down. That is the definition of a liquidity crunch: at a given shock point, people want money more than other less-liquid assets. In this model, the Pb "shock" should be to increase the price of the medium of exchange, not halve it. The internal story doesn't make sense, even if the end results "seem" right.

(Speaking of results: "B-owners"/agents-with-excess-liquidity do poorly in this model but very well in real-world AD depressions: this model recommends minimizing the amount of medium-of-exchange pre-shock. Compare B's final consumption bundle - "50 A's, 100 B's, and 50 C's" - with A's - "200 A's, 100B's, and 50 C". That's not the final result I see in real life if B is money and A is a non-liquid asset.)

Thinking a little more on this: you also get a medium-of-exchange failure if the "shock" occurs on the utility function. If U=log(A)+log(B)+log(C) becomes U=log(A)+2log(B)+log(C) (ie, the utility value of money doubles but sticky prices inhibit transmission of that shock to the market in the short/1-time-period term), you get a monetary failure that looks much more like an AD-induced depression:

Those with an initial endowment of 300 B end up with 75 A, 150 B, and 75 C, a utility "value" of 8.1.

Using a non-barter system:

Those with an initial endowment of 300 A end up with 200 A, 75 B, and 25 C, a utility "value" of 7.4. The same value applies to the initial endowment for C.

And with barter:

Both A and C initial endowments maximize at 112.5 A, 75 B, and 112.5 C, a "value" of 7.85 (still worse than the initial endowment of liquidity but better than the non-barter system).

I think this story - a shift in utility preference coupled with sticky prices - makes more sense when explaining AD recessions in a medium-of-exchange system.

I don't have my own blog though, so I could be totally off the mark. :)

Kevin: it's sunny here too. Went urban canoeing today, and worked on my daughter's Tercel. My brains not totally on economics either!

OK, I now get what you are saying. I haven't totally wrapped my head around your 3D picture yet.

Roger, assume competitive equilibrium, but prices are sticky. See my reply to Max below.

Max: you are not totally off the mark.

Here's one way to think of it: B is also the unit of account (the prices of A and C are measured in units of B, so Pb=1 by definition). Initially the endowments are 300 A, 600 B, and 300 C, and prices are at market-clearing equilibrium with Pa=Pc=2. Then suddenly the endowment of B falls to 300, but prices are sticky, so Pa and Pc stay at 2. We then get exactly the sort of recession I have described.

Or we could have a preference shift, like you have described, with sticky prices.

Pithily, a 3-good, 3-agent Edgeworth box is 3 orthogonal 2-simplexes, i.e. triangles. (A 1-simplex is a segment, 2-simplex is a triangle, 3-simplex is a tetrahedron, etc. And you need a (n - 1)-simplex to show the allocation of one good among n agents, such that all quantities will sum to a constant.) A triangle is 2D, so this is indeed a 6-dimensional construction.

Pithily, a 3-good, 3-agent Edgeworth box is 3 orthogonal 2-simplexes, i.e. triangles. (A 1-simplex is a segment, 2-simplex is a triangle, 3-simplex is a tetrahedron, etc. And you need a (n - 1)-simplex to show the allocation of one good among n agents, such that all quantities will sum to a constant.) A triangle is 2D, so this is indeed a 6-dimensional construction.

Pithily, a 3-good, 3-agent Edgeworth box is 3 orthogonal 2-simplexes, i.e. triangles. (A 1-simplex is a segment, 2-simplex is a triangle, 3-simplex is a tetrahedron, etc. And you need a (n - 1)-simplex to show the allocation of one good among n agents, such that all quantities will sum to a constant.) A triangle is 2D, so this is indeed a 6-dimensional construction.

anon: thanks. I wish I could visualise it!

What would this model predict for an "inflationary" environment, i.e. Pa = Pc = .5Pb?

John: lovely question! I should have seen that one coming, but I didn't.

It's late, and my brain is too slow and tired to do the math. But I think we will also get too little trade in A and C. But this time we will have an excess demand for A and C, and the (unenforceable) barter deal the A-owners and C-owners would like to do is "I will sell you more of my A, but only if you agree to sell me more of your C in return."

(I *think* it's the same consumption vector as in my example above where Pa=Pc=2Pb, but I'm not sure, and it would depend on the particular rationing scheme, because both the B-owners and C-owners would want to buy more A, and both the A-owners and B-owners would like to buy more C.)

John: It's late, and I may have got this wrong:

*One* solution is:

A-owners consume 240 A's, 120 B's, 0 C's.
B-owners consume 60 A's, 60 B's, 60 C's.
C-owners consume 0 A's, 120 B's, 240 C's.

The A-owners don't want to sell any more A's for B.
The C-owners don't want to sell any more C's for B.

A barter deal would give them infinite marginal benefits.

Weird, huh?

John, I think "Pa = Pc = .5Pb" is a deflationary environment: good B is the money/unit of account and it goes from buying 1 unit of goods A or C (when Pa = Pc = Pb) to 2 units. The money has gained purchasing power: the opposite of inflation. Nick's initial model describes an inflationary environment, with the amount of money required to buy a given unit of A or C doubling (Pa=Pc=Pb -> Pa=Pc=2Pb). That's why actors who started with 300 units of B (money) have lower equilibrium utility on the equilibrium even though the market for currency still clears. Right?

Nick, I think the math is wrong on "Pa = Pc = .5Pb":

B-owners maximize utility at 200 A, 100 B, 200 C, trading 100 units of B for 200 units of A and C respectively.

A (and, in reverse, C) owners then maximize with a 100 A (or C) and 100 B, which becomes 50 A, 100 B, and 50 C, as A "buys" 50 units of C with 25 units of B (and C does the same).

Also Nick, thanks for the explanation (on the endowment of B falling from 600 to 300 while prices stay high).

Simply put, very good.

If the Pb > Pa = Pc the solution is IMO always structured in a way that Bs can optimize their position unconditionally but the others are conditional on how much they are able to sell (because money is short).

But if Pb <= Pa = Pc there is no constraints and clearing happens? So "Pa = Pc = .5Pb" solution would be for B: 4/3*100 A, 4/3*100 B, 4/3*100 C and for the rest: B: 2/3*100 A, 2/3*100 B, 2/3*100 C (Bs net worth is double to start with)? E.g. while Max's solution for B is the best B can achieve there is no logical reason to assume the others are happy to trade more than 4/3*100 unit of their goods.

I hate there is no edit. The above solution in the last paragraph should be:

B: 150 A, 150 B and 150 C (double net worth)

and for

A&C: 75 A, 75 B and 75 C

(Now A+B+C == 75+150+75 == 300)

Max: John calls it "inflation" (Barro and Grossman called it "repressed inflation") because the price of non-money goods is lower than equilibrium, so there is upward pressure on the price of non-money goods, and so the price of non-money goods will eventually rise (relative to the money-good).

Jussi: "But if Pb <= Pa = Pc there is no constraints and clearing happens?"

I think you typod and meant Pb > = Pa=Pc

No. There is excess demand for A and C, so the a and b people won't be able to buy as much C as they want, and the b and c people won't be able to buy as much A as they want.

John: yes, I think my 12:24AM solution is correct, but it is only one solution, and there is a continuum of other solutions. My solution assumes that the B-owners do all their shopping first, and buy all the A's the A-owners want to sell, and buy all the C's the C-owners want to sell, before the A-owners and C-owners can start shopping to buy each others' goods.

Yes, typo, thanks.

But Nick's:

"*One* solution is:

A-owners consume 240 A's, 120 B's, 0 C's.
B-owners consume 60 A's, 60 B's, 60 C's.
C-owners consume 0 A's, 120 B's, 240 C's."

for "Pa = Pc = .5Pb" cannot be right as B starts with more net worth and ends up with less. In money terms (Pb) B starts with 300 and ends up with 30+60+30 = 120.

I *think* still not having time to write the steps down that B is willing and *able* ("got money") to trade until A=B=C=150 (== B's utility max). For symmetry reasons then other got A=B=C=75? Or am I mistaken?

Jussi: people maximise their utility, not their net worth.

I was trying to do accounting not optimization. What Pa = Pc = 0.5Pb then means - if the money worth is changing during trading someone is not using the prices given. So why B's net worth is first 300 but valued after trades only at 120 (30+60+30)?

Is there anything in the model that would stop the following set of trades leading to the same outcome from monetary exchange as the barter outcome (its early for me so hope my logic is valid) ?

Nick: I just dont see how...

"*One* solution is:

A-owners consume 240 A's, 120 B's, 0 C's.
B-owners consume 60 A's, 60 B's, 60 C's.
C-owners consume 0 A's, 120 B's, 240 C's."

...could be right at all. Even aside from the dissapearing purchasing power of B-owners (did the price of B change?), A and C owners could maximize utility by trading 60 units of B for 120 units of the other good. That results in a 120, 120, 120 final bundle for both A and C initial endowments (though I think that's wrong: for the problems with B I identified above).

Jussi:

You're right on the final consumption bundle. My assumption was that the buyers in this model (the agent transforming B into A or C) had no agency: B would maximize utility given its initial endowment and the A and C endowments would maximize utility given their secondary share of B.

With symmetric agency (and the same two step process) it ends your way: B trades 75 B for 150 A and 75 B for 150 B. This maximizes step one utility at A/C (150,75,0) and B (150,150,150). A and C then trade 32.5 units of B for the opposite good, maximizing utility at (75,75,75).

Max, yes, I was also starting with conditional optimization. But then after reading comments (like "*one* solution") I realized it all depends on order of execution. But if A and C are rational they will play it in a way that they max it out too.

I think that trade reduces the utility of the C-types.

Max: "Even aside from the dissapearing purchasing power of B-owners (did the price of B change?)"

No. B-owners trade 240 B's for 60 A's plus 60 C's, so their budget constraint is satisfied if Pa=Pc=2Pb.

"A and C owners could maximize utility by trading 60 units of B for 120 units of the other good."

Only in a barter deal. A-owners can't find anyone who wants to sell more C for B, and C-owners can't find anyone who wants to sell more A for B.

Nick,

isn't "3. 100 B for 50 A (by those who start with C with those who start with A)" one of the trades that takes place in your model to reach the "300 A's consume 200 A's, 100B's, and 50 C's" outcome ?

I assumed that they traded the 100 B's they got for selling their 50A to the B-endowed, with the C-endowed . At that point the C-endowed have 200B (100 from trades with the B-endowed and 100 from trades with the A-endowed).

The C-endowed can then trade 150 of those B for 75A's (rather than 100B's for 50As as happens in my understanding of your model). Perhaps I am misunderstanding the utility function, but even if this was the last trade wouldn't the C-endowed be better off (50B, 75A,200C v 100B ,50A,200C) with this trade? And the A-endowed would take the trade a as long as they were optimistic they could offload some of their B's (they would have 250 at that point) for additional C's (and indeed the C-endowed will happily make that trade.

MF: this is not part of the model: "were optimistic they". Agents trade only on utility improving basis.

Nick: you are talking about the original constraint. Max and I were discussing "0.5" case. In that case money is too valuable and it gets traded away when there is excess of it.

Jussi,

"Agents trade only on utility improving basis."

Fair enough. But then B is not much of medium of exchange if the A-endowed don't want to sell A's for money (B's) because they fear they will not be able to trade that money for other goods.

MF, yes, not much because the (medium of exchange) price was wrong.

MF: that fear will be sometimes be rational if there is excess demand or excess supply of those other goods. And we assume large numbers of agents, so no individual's trades can affect his expectations about whether there is excess demand or supply of those other goods.

Mick,

True. But relating this model closer to the real world:

People with money buy services that cause all plumbers and electricians to earn \$100 each. The plumbers use this \$100 to buy services from the electrician. But when the electricians (who now have \$200 cash) offer to buy \$150 dollars of services from the plumbers they refuse the deal because they are not sure they will be able to spend the money in a way that increases their utility.

Is that what happens in a recession where plumbers feel underemplyed?

Apologies, Nick (not Mick).

If the plumber buys an extra \$100 of labour from the electrician, the plumber knows it is very unlikely the electrician will spend an extra \$100 to buy labour from that particular plumber.

I'm almost certainly missing something obvious here but if the plumber buys an extra \$100 of labour from the electrician its because he values the extra electrical work more than the money even if he has no expectation of selling more of his own labour as a result.

But if (by assumption) all electricians now have an extra \$100 as a result of the extra spending by plumbers, and they want to spend some of this money on plumbing services (as its the only other thing in the economy!) then why would the plumbers (observing that its a buyers market and feeling underemployed since its a recession) not be keen to swap plumbing services for (undervalued) money?

For what its worth I figured out what I was missing.

Its not that the plumbers won't sell plumbing services if offered - its that the electricians wouldn't offer to buy any more at the current price level. (Or in the original model, at the point where the A-endowed have 150 A's, 100 B's, and 50 C's it would reduce their utility to buy more C's at 2Bs each).

Nick said: "Why do you think I wrote down the utility function!"

I said: "I'm probably going to run afoul of some assumptions"

Do whatever needs to be done so the demand of A is 70 each, the demand for C is 70 each, the endowment of A is 300 total, and the endowment of C is 300 total.

Consider the case where P(A) = P(C) = 2P(B). Since each exchange between B's and A's or C's will involve two B's, let us exchange two B's for one D. Then P(A) = P(C) = P(D). Each person who started with 300 B's now starts with 150 D's. U = log(A) + log(B) + log(C) = log(A) + log(2D) + log(C) = log(A) + log(D) + log(2) + log(C). Since log(2) is a constant, maximizing log(A) + log(D) + log(C) will also maximize log(A) + log(B) + log(C). People will act as though P(A) = P(B) = P(C) and the people with only B's start with 150 B's instead of 300.

Nick Rowe: "It's got nothing to do with "too much saving". . . .

"It's got nothing to do with the real interest rate being wrong. . . .

"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."

Or, as the case with 150 B's indicates, there is not enough money.

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