A simple question about Walras Law

Inspired by Free Radical's post, I think I have figured out a simpler and clearer way to say what I want to say about Walras' Law.

Ask yourself the following question:

Q. Assume an economy where there are (say) 7 markets. Suppose 6 of those markets are in equilibrium (with quantity demanded equal to quantity supplied). Is it necessarily true that the 7th market must also be in equilibrium (with quantity demanded equal to quantity supplied)?

This is an open book exam, and you may Google if you wish.

This is a question for both macroeconomists and microeconomists. (And for non-economists too.)

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My answer is "no".

Let there be 8 goods: money; and 7 others (apples, bananas, carrots, dates, eggs, figs, and grapes). There is: a market in which apples are bought and sold for money (the "apple market"); a market in which bananas are bought and sold for money (the "banana market");....and a market in which grapes are bought and sold for money (the "grape market"). Which makes 7 markets in all. And let each of those 7 foods have a price in terms of money. And let 6 of those 7 prices be perfectly flexible, and adjust to equalise quantity demanded and quantity supplied in a perfectly competitive market. But let the price of grapes be fixed by law.

[Update: in response to Arnold Kling. I mean the price of grapes is fixed by law in the same way that the price (rent) of rent controlled apartments is fixed by law. By making it illegal to buy or sell grapes at any price above or below $1 per kg. And not by the central bank buying and selling grapes at a fixed price to create a "grape standard" monetary system.]

It is perfectly possible to have an excess supply of grapes (if the price of grapes is fixed too high), even if all 6 other markets have quantity demanded equal to quantity supplied.

The "grape market" is a market where grapes are exchanged for money. If there is an excess supply of grapes, it means people want to sell more grapes than people want to buy. So some people will be unable to sell as many grapes as they want to sell. And that means that those same people are unable to "buy" as much money with grapes as they want to buy. If the price of grapes is fixed by law at $1 per kg, and there is an excess supply of 100kgs of grapes, and so an excess supply of $100 worth of grapes, then there is also a $100 excess demand for money in the market for grapes.

In any one of those 7 markets, at any price (whether fixed by law or not) the value of the excess supply (demand) of apples (or whatever) must equal the excess demand (supply) for money in that same market. Simply because people plan on paying (and getting paid) for the stuff they buy (and sell).

And that same thing holds true in a barter economy, where there is no money. If there is a market where apples and bananas can be swapped (the "apple-banana market") the value of the excess supply (demand) for apples in that market must equal the value of the excess demand (supply) of bananas in that market.

And that same thing holds true in a market where arbitrary combinations of 3 or more goods can be swapped for arbitrary combinations of those same 3 or more goods. ("Can I swap a basket of 10 apples and 5 bananas for the same value of carrots?"). The values of the excess demands must sum to zero in that particular market.

And the same thing holds true in an economy where there is only one market where arbitrary combinations of all goods can be swapped for arbitrary of all goods. The values of the excess demands must sum to zero in that one market.

And only in that last case, an economy with only one market where all goods are traded for all goods, is it true that the values of the excess demands of all goods must sum to zero for the whole economy.

Walras' Law is true and useful for the economy as a whole only if there is only one market in the whole economy, where all goods are traded for all goods. But that is not a monetary economy. And it is not a real world economy.

In a monetary exchange economy, where every other good is bought and sold for only one good, that we call "money", there are as many different excess demands (supplies) for money as there are markets. There can be an excess demand for money in the apple market, and an excess supply of money in the banana market.

Is that what you thought? Is that what you were taught? Is that what you have read about Walras' Law?

And does that make it simpler and clearer?

Comments

[Arnold Kling]

Not at all.

How does the price of grapes get fixed? If the government does it by buying and selling grapes at the fixed price, then we are on a "grape standard," and with flexible prices all markets clear.

This post did not convince me of anything, and it left me confused.

--Arnold Kling

[Nick Rowe]

Arnold: Damn! then I have failed!

What I meant was: the price of grapes is fixed by law, in the same way that the price (rents) of rent-controlled apartments is fixed by law. Not by the government renting or letting apartments.

[Market Fiscalist]

It seems easy to imagine a situation where 6 markets are in equilibrium and one is not.

but:

- If I plan to buy grapes but can't I will have excess money. Won't I take that excess money to the other 6 markets and disrupt them ?
- If one of the other goods has perfectly flexible prices won't its price adjust to soak up any overall total excess/deficit of money?

So: I'm going with "no". (I'll probably regret it)

[Matthew]

I don't think this is quite right.

We have excess supply in the grape market, but supply=demand in all other markets. The seller of grapes is expecting to sell all of his grapes. Now either his demand functions for each of the other goods is computed including the income from those unsold grapes or not. If not, then the individual has planned expenditures that are less than income px < w, and thus failed to maximize utility, hence no walras law. If he did, then supply=demand in all these other markets means that the grape seller has actually planned more expenditures than he has wealth, so px > w, hence no walras law.

If you stipulate that px=w and local non-satiation, walras law holds.

[Matthew]

Argh. HTML chopped off my text because I tried to use inequalities. [I fixed it. You need to put a space either side of < or > .NR] meant to say:

We have excess supply in the grape market, but supply=demand in all other markets. The seller of grapes is expecting to sell all of his grapes. Now either his demand functions for each of the other goods is computed including the income from those unsold grapes or not. If not, then the individual has planned expenditures that are less than income px less than w, hence no walras law. Thus, planned expenditures in all the other markets is based on income including those unsold grapes, so that without that income we now violate the grape seller's budget constraint.

If you stipulate that px=w and local non-satiation, walras law holds.

[Tom Brown]

Nick, interesting post. I only made it through the first half, but let me say right now I always suspected there was something fishy about an economy with seven markets! Thanks for confirming that. :D

[Nick Rowe]

Matthew: "Now either his demand functions for each of the other goods is computed including the income from those unsold grapes or not. If not, then the individual has planned expenditures that are less than income px < w, and thus failed to maximize utility, hence no walras law."

He maximised utility **subject to the additional constraint that he was not able to sell as many grapes as he wanted to sell**. Which may affect his demands for goods in the other 6 markets. But the prices in those other 6 markets (by assumption) adjust to clear those markets. But he continues to *want to sell* more grapes than he actually sells, so there is an excess supply of grapes.

Plus, by the way, money is one of the goods in the vector x. But there are 7 excess demands for money - one for each of the 7 markets in which money is traded. Even if 6 of those excess demands for money are zero, the seventh doesn't need to be zero.

MF: "- If I plan to buy grapes but can't I will have excess money. Won't I take that excess money to the other 6 markets and disrupt them ?"

Very probably yes.

"- If one of the other goods has perfectly flexible prices won't its price adjust to soak up any overall total excess/deficit of money?"

It's price will adjust to soak up any excess supply or demand for money **in that particular market**. But there remains an excess supply of money **in the grape market** (if there is an excess demand for grapes, so people can't buy as many grapes as they want).

[Market Fiscalist]

"Very probably yes"

Doesn't this mean that if the 7th market being out of equilibrium disrupts the other 6 then the only way that will not happen is that the 7th must also be in equilibrium ?

Also: If my desire to buy banana's is frustrated and I use the excess money to buy more apples (or just decide to hold more cash) isn't the excess supply of money in the banana market gone ?

[Geez]

I suggest you all go back to your graduate school notes and look at the proof of Walras' theorem.

[Nick Rowe]

Geez: Max U(x) st p.x = p.e (assuming non-satiation for equality, and an endowment economy for simplicity)

therefore p(x-e)=0 for each individual.

Add up across all agents: p(X-E)=0.

Simple, trivial, and wrong.

[Nick Rowe]

What *should* have been taught in grad skool:

Let there be n goods. So there are n-1 markets (assuming a monetary exchange economy).

For each of those n-1 markets:

Max U(x) st p.x=pe *AND st any quantity constraints on the amount you can buy or sell in any of the **remaining n-2 markets** (because some of those n-2 other markets may have excess demand, where buyers are rationed in how much they can buy, or excess supply, where sellers are rationed in how much they can sell).

That means there are (up to) n-1 different decisions, each subject to different constraints. If you add up the excess demands from n-1 different decisions, you will not get Walras' Law.

Benassy's "multiple decision hypothesis" IIRC. (A generalisation of Clower's "dual decision hypothesis").

If they didn't teach you that in grad skool, then they taught you wrong.

You young whippersnappers think you know everything. But the discipline of economics has regressed, in some respects. Dark Age, etc.

Market Fiscalist is doing best so far, because he at least gets the idea of spillovers due to quantity constraints.

If you can't buy as much butter as you want (because there is excess demand in the butter market) you might demand more margarine than you would otherwise want to buy in the margarine market. But that doesn't stop you wanting to buy more butter.

If you can't sell as much labour as you want to sell (because there is excess supply in the labour market) you might demand less butter than you would otherwise want to buy in the butter market. But that doesn't stop you wanting to sell more labour. Keynes.

[Nick Rowe]

MF: "Doesn't this mean that if the 7th market being out of equilibrium disrupts the other 6 then the only way that will not happen is that the 7th must also be in equilibrium ?"

Yes.

"Also: If my desire to buy banana's is frustrated and I use the excess money to buy more apples (or just decide to hold more cash) isn't the excess supply of money in the banana market gone ?"

No. You still want to buy more bananas than you can (and sell more money in the banana market than you can). If someone asks you "Do you want to buy more bananas for money?" you answer "Yes!".

[Nick Rowe]

MF: you are thinking about a world where the banana market is only open on Mondays, and the apple market is only open on tuesdays, etc. So if someone asks you on wednesday if you want to buy more bananas, you *might* say "No, too late! I've already spent all my cash, and bought enough fruit for the week".

[Market Fiscalist]

OK, if I have a constant stream of income and always have a greater desire for bananas at the current price than I can obtain , then I agree it is reasonable to say their is an ongoing excess supply of money and under-supply of bananas in the banana market.

[Nick Rowe]

MF: you got it.

[Jason]

Hi Nick,

Did you just re-derive the sub-additivity of joint entropy?

I arrive at the same conclusions as you using information theory. Essentially adding together the markets doesn't necessarily tell you about the other markets because of the mutual information (prices of 2 or more good combinations) unless you say they're all independent.

http://informationtransfereconomics.blogspot.com/2014/08/walras-law-information-theory-edition.html

[Matthew]

Nick: so is your point just that excess demand/supply spills over into other markets? I think I might be inclined to agree, as Walras Law is a statement about Walrasian Equilibrium, not a prediction about disequilibrium behavior.

Maybe Im not understanding the grape example, so let's try another one. In the real world there are industries that are unionized and industries that are not unionized, so let's assume there's a union labor market and a non-union labor market. The labor union has bargained a wage floor above the market clearing wage so that there is now excess supply of labor in the union labor market. Realistically, what happens here is workers apply for a higher-paying union job first, and those who fail to get one then apply to a non-union job. There's no wage floor in the non-union labor market, so the wage falls to accommodate all of the workers rejected from union jobs (who still want to work at the lower wage).

I can see how you could argue that the non-union market is in equilibrium while the union market is the only market that is not in equilibrium. From the perspective of Walras' Law, however, both the union and non-union labor markets are in disequilibrium, because spillovers from the union to non-union market have based their decision on information other than price, which violates the definition of Walrasian equilibrium. Hence, I could agree that Walras' Law is not a useful principle in practice.

Would you say that's more or less the same as the point you're arguing? Or have I misunderstood?

[Frances Woolley]

Can I apply for a deferred exam?

[Nick Rowe]

Matthew: I think the spillovers thing is an important part of it, but it's not the main thing.

If there are n goods, how many excess demands are there?

In Walrasian GE theory, there is one market where all n goods are traded, so there are n excess demands. One excess demand per good.

And the values of those n excess demands in that market must sum to zero.

In a monetary exchange economy, where one of those n goods is used as the medium of exchange, there are n-1 markets, and in each of those markets 2 goods are traded: money, and one of the other goods. So there are 2(n-1) excess demands. There is one excess demand for each of the non-money goods, and n-1 excess demands for money.

And in each of those markets, the values of those 2 excess demands in that market must sum to zero.

[Nick Rowe]

Jason: I read your post, but I'm afraid I don't understand it.

Frances: of course!

[Tom Brown]

Nick: I read (all of) your post, and (amazingly enough) I'm afraid I *do* understand it. (... and I'm this close ->||<- to understanding Jason's too). Not typical for me by any means.

[Tom Brown]

Nick you write:

"MF: you are thinking about a world where the banana market is only open on Mondays, and the apple market is only open on tuesdays, etc. So if someone asks you on wednesday if you want to buy more bananas, you *might* say "No, too late! I've already spent all my cash, and bought enough fruit for the week"."

Does that imply we can add some timing constraints to a multi-market economy and devise a way to resuscitate Walras' Law for a multi-market economy ?

[Nick Edmonds]

When I think of Walras Law, I think of a market to mean the quantity demanded and the quantity of supplied of a particular good, given the set of prices. Nothing more. Certainly nothing to do with what other goods or money might be offered in exchange. So for n goods, you need n markets. You also need a budget constraint. So given the initial endowments (including money holdings), the set of prices and the budget constraint, everyone decides how much they want of each good (and closing money holdings). In this model, people only care about the end holdings - they don't care about whether they exchange apples for bananas or apples for money or bananas for money, whatever.

So for this purpose, each "market" is just the quantity demanded and the quantity supplied, and you need n markets for n goods. If you want to use the concept of a "market" as something slightly different, then you're maybe going to get a different result.

[Nick Rowe]

Nick E: let there be n goods. (Money, if it exists, is one of those n goods.)

In a Walrasian economy, there is one market, where each of the n goods is traded against any of the other goods, and each good has one demand and one supply, and so one excess demand. There are n excess demands. That is the world you are thinking of. Yep, Walras' Law works in that world (provided you are talking about *notional* excess demands, which are the excess demands people would choose if they assumed they faced no quantity constraints of how much they can buy and sell).

In a monetary economy, there are n-1 markets, with 2 goods traded in each market. Each good has one excess demand, except money, which has n-1 excess demands. (Money can be in excess demand in the apple market, and in excess supply in the banana market).

In a barter economy, there are n(n-1)/2 markets, and each good has n-1 excess demands.

Tom: "Does that imply we can add some timing constraints to a multi-market economy and devise a way to resuscitate Walras' Law for a multi-market economy ?"

No. If you have time in the model, you would need to assume there is one big market that opens at the beginning of time, then closes forever.

[Mike Freimuth]

Nick,

For the record, I think I have always understood your point and I have always agreed with it, I just think it goes to far to declare Walras' Law useless. It's meant to describe a certain market mechanism in which prices adjust to clear markets. I don't think it's supposed to be a theory of cross-market effects of price controls (or sticky prices or whatever), or at least it shouldn't be. If you want to model a sticky price in some market, for whatever reason, then you have to change the model by putting a quantity constraint in somebody's maximization problem and then, yes, the other markets can all clear.

I think there are two separate issues here though. The above should be true even in an n-good economy with n markets where each market trades for money and money is not treated as a good but merely a way of measuring incomes and expenditures. But this would not be a general glut, it would just be a defect in some particular market that caused no particular problem for the economy as a whole. This still leaves the general glut unexplained. I think (though I could be wrong) that we would agree that a general glut is the result of excess demand for money in all (or at least many) markets, but this raises a different and more important question which is: why would people "demand" money as such (as opposed to money just representing other goods). I think that is the main issue, not how many markets there are.

Obviously, I have my somewhat novel explanation for why this is but just let me point out that I think this highlights the absurdity of the pure liquidity theory which seems to imply that people have an excess demand for "liquidity" precisely at times when nobody wants to buy anything.

[Mike Freimuth]

BTW, I think I kind of agree with you about economics regressing. I got an intro text for a class I may or may not be teaching at a community college and it is tragically infected with cardinal utility. I blame the internet....

[Roger Sparks]

"Let there be 8 goods: money; and 7 others .......... And let 6 of those 7 prices be perfectly flexible, and adjust to equalise quantity demanded and quantity supplied in a perfectly competitive market. But let the price of grapes be fixed by law."

OK, I think we can have a Walrasian auction here for 6 items. The auction will adjust the price to a level where demand equals supply resulting in a 6 clearing markets.

The price controlled market has not yet been considered. Should we consider it before holding the Walrasian auction or after?

I think the timing (of the controlled market trade) could affect the other markets by influencing the amount of money available BUT that possibility also introduces a potential limitation on the 8th good (which is "money"). Maybe the quantity of money is important, just as the amount of goods for each of the other markets is important.

It seems to me, if the Walras' law is to hold, that ALL the money available to trade in the market must also trade, just as all the goods must trade. Thus, at market close, every good has exchanged at some price, and the price-times-goods demanded would equal the price-times-goods supplied.

I would conclude that the act of government control of one price would have distorted the market. The effect of the distortion would depend upon the sequential order of market price setting.

[Mike Freimuth]

Roger,

It's not the timing that matters, it's whether market participants take the constraint into account when choosing the values of other goods to consume. One way to think of it is to say that the market for the price-controlled good happens first, but if it happens after and people know what is going to happen in the first place, it will be the same. Trying to mess with the timing of the market produces further problems with the framework of the model. For instance, income in the model comes from the sale of goods so how do people use their income from the good that gets sold last? You can come up with some kind of answer to that if you want but if you try, you will end up with a completely different model.

Regarding this: " if the Walras' law is to hold, that ALL the money available to trade in the market must also trade, just as all the goods must trade."

That highlights my point still further. There is no actual money in the model (even the version with money). What I mean is that there is not a particular quantity of money or a velocity or a price level (the equilibrium you get is really a vector of relative prices). "Money" is only used to *measure* values of prices and incomes. This allows you to represent a frictionless barter economy for n goods with only n prices rather than n(n-1)/2 prices. That is a lot different from modeling the actual function of money in the economy! Doing so is much more complicated.

[Nick Edmonds]

Nick,

None of your n-1 markets alone tells you what the actual demand for money balances is. To get that you have to aggregate all n-1 markets. What you get is the total demand and supply of money. That's your nth market.

You might say that this nth market is nothing more than the other n-1 markets added up and I'd agree, but to me that's all that Walras Law is saying.

[Nick Rowe]

Sorry for not responding. I've spent too long reading about Rotherham.

Nick E: There are n-1 excess demands for money. When you add them all up, what do you get? Is it useful?

Suppose I have $10 in my pocket, and want to spend it all on fruit. I want to buy apples, but they have run out. $10 excess demand for apples, and $10 excess supply of money in the apple market. So I try to buy bananas, but they have run out of those too. So there's another $10 excess supply of money in the banana market. That's $20 excess supply of money in total. Add more types of fruit to the story, and it becomes even more ridiculous.

Let A be the excess demand for apples, and let a be the excess demand for money in the apple market.

Then A+a=0 (call that "Nick's Law" because it's trivial and obvious.)

Similarly, let b be the excess demand for bananas, and b be the excess demand for money in the banana market.

So B+b=0

Etc for carrots and dates and.....

Adding them all up we get

A+a+B+b+C+c+......=0

Rearranging, we get:

A+B+C+......+a+b+c+.....=0 (Walras' Law, true version)

Whoopydoo! Yes! 0+0+0+0+0+......=0

If you add up a whole load of zeros, you get zero!

But Walras' Law, true version, tells you absolutely nothing that the totally trivial "Nick's Law" doesn't tell you! It gives you no cross-market restrictions.

[Nick Rowe]

Mike: Generally agree with your main point. Non-market-clearing prices will cause spillovers whether we have a monetary economy, barter, or a single Walrasian market. But the nature of those spillovers will be very different, depending on what sort of markets we have. For example, if we start with a monetary economy, and an excess supply for both labour and output, opening up a barter market, where labour can be swapped directly for output, would make a big difference. Employment and output would increase.

"but this raises a different and more important question which is: why would people "demand" money as such (as opposed to money just representing other goods). I think that is the main issue, not how many markets there are."

Because holding a *stock* of money (an inventory of money) is very useful for buying and selling things. It is very very hard to spend money one instant after you get it. That's the only way you would not hold a stock of money on average.

[Nick Rowe]

Roger. "The effect of the distortion would depend upon the sequential order of market price setting."

Good point. But Mike's response is good too. The sequential order may affect expectations of those constraints, but it's the expectations that matter.

[Nick Edmonds]

Put another way, each person's budget constraint implies that their planned net expenditure less planned change in money balances sums to zero. Sum them all up (still equal to zero) and rearrange to get Walras law. I don't disagree that it's trivial, but that's what I think it is.

[Nick Rowe]

Nick E: But consider my example where I plan to spend $10 on apples, can't find any apples so plan to spend $10 on bananas instead, and can't find any bananas. We have a $10 excess demand for apples, plus a $10 excess demand for bananas, so that's $20 excess demand for fruit. What is my excess supply of money? $10, or $20? You need it to be $20 for Walras' Law to be true. But I only have $10 in my pocket, and have no intention of holding a negative balance..

[Majromax]

> But consider my example where I plan to spend $10 on apples, can't find any apples so plan to spend $10 on bananas instead, and can't find any bananas. We have a $10 excess demand for apples, plus a $10 excess demand for bananas, so that's $20 excess demand for fruit.

I don't think that's a well-formed problem.

The excess demand for bananas is intrinsic, but the excess demand for apples depends on there also not being any bananas. Your non-cleared market would clear with any linear combination of the following (summing up to 100%):

* You have $10 less to spend
* There are $10 more bananas on the market
* There are $10 more apples on the market

This is where the formulation of Walaras' law gets into trouble, because it isn't built to express a conditional shortage. This isn't a problem with a centralized N-good market when there aren't any conditional decisions to be made, but it becomes a problem when we look at pairs of goods (money and something else) in isolation.

This isn't a purely abstract problem, either. It's important for the apple-growers to know that if the banana supply recovers (or banana prices go up) then their seemingly excess demand will disappear.

[Nick Edmonds]

I'm sure you know this better than I, but it doesn't sound right to me that there is $20 of excess demand for fruit there. For a start that's not consistent with the budget constraint. More conceptually, you certainly only want an extra $10 of fruit. You want $10 of apples OR $10 of bananas and you'll take the apples in preference, but you don't want $10 of apples AND $10 of bananas. I'm not how you determine the excess demand in that sense, but adding it doesn't seem right.

[Nick Rowe]

Majromax: we are on the same page (except you switched apples and bananas in my example!)

But suppose we are an outsider observer, who doesn't know preferences, and which demands are conditional and which not. All we actually observe are: excess demand for apples, and excess demand for bananas. Where's the good which is in excess supply? We know there must be an excess supply of money in the apple market, without even looking at the banana market. We know there must be an excess supply of money in the banana market, without even looking at the apple market.

What does Walras' Law tell us that we didn't already know from "Nick's Law"?

"I don't think that's a well-formed problem."

It isn't *one* well-formed problem. It is two *different* well-formed problems. But because they are different problems, when you add them together you get nonsense.

Assume (to keep it simple and symmettric) both markets are open at the same time, and I know they are both out of fruit.

Apple market problem: choose A to max U(A,B) s.t pa.A=pb.B=$10 AND s.t. B=0

Banana market problem: choose B to max U(A,B) s.t pa.A=pb.B=$10 AND s.t. A=0

In each market, we max U st the regular budget constraint plus st quantity constraints in the OTHER market(s).

Nick E: see my answer to MajroMax above. An outside observer, who doesn't know preferences or anything, only observes the excess demands in the markets. How can he possibly know whether it is or is not legitimate to add them up? Walras' Law is no help.

[J.V. Dubois]

Very nice blogpost. However what I find intriguing here is why people say (like Free Radical post you linked to) that says: "I don’t think this model was ever intended to be a model of disequilibrium."

I find this strange since the very definition of "excess demand" is what defines disequilibrium. Let's take your example of grapes for $1 per kg. Let's assume that I have no need for grapes at that price (I buy apples instead) but it may change if they are for $0.90. So at all times I demand some amount of grapes if they are at this lower price. However it is the "excess demand" for grapes at this price that pushes it to its equilibrium price of $1 in the first place!

So in the same vein it is strange to talk about "excess demand" for safe assets. Why should be there an excess demand for those? Is their price sticky so that excess demand cannot be eliminated by change in price?

[Nick Rowe]

JV: Thanks!

"I find this strange since the very definition of "excess demand" is what defines disequilibrium."

Yep. Robert Clower said something similar, IIRC. The sort of stability analysis that gets done using Walras' Law is useless.

On safe assets: yep. I think what they should be saying is "excessive demand for safe assets". It's not an "excess demand" in the way we normally use those words.

[Mike Freimuth]

JV,

I tried to clarify this in a comment but any model of equilibrium has to say something about what happens if you aren't at an equilibrium. My point is that the point of the model is to say that if you have an off-equilibrium price vector then you have excess supply in some market and excess demand in some other market *and therefore, those prices should adjust to bring those markets into equilibrium and this will lead to a general equilibrium.* It's not meant to describe what would happen if certain prices can't adjust but people know this and change their behavior in other markets accordingly. That is a different mechanism and a different model.

[Seamus Hogan]

Nick,

I don't think this is right. It is hard for an accounting identity to be false, so there must be some slippage in usage. I think the slippage comes when you said there I are 7 markets. But the conventional use of Walras' law would say there are 8 markets (7 for goods and 1 for money). The fact that transactions only operate pairwise through money is irrelevant here. The point about Walras' Law in GE is that we only have n-1 independent relative prices (and so can arbirtrarily make one of the goods the numeraire as you have with money here), but also only n-1 independent S=D conditions. Walras' Law in your monetary economy says that if S=D for each of the non-money goods, then we don't need to worry about checking if S=D in the 8th market (money). By restricting your attention to only the 7 non-numeraire markets in the first place, you were implicitly invoking Walras' Law from the outset.

[Nick Rowe]

Seamus: whether or not there's a numeraire, and whether or not that numeraire is also the medium of exchange, is a quite separate question.

It's about what markets exist. You say: "Walras' Law in your monetary economy says that if S=D for each of the non-money goods, then we don't need to worry about checking if S=D in Walras' Law in your monetary economy says that if S=D for each of the non-money goods, then we don't need to worry about checking if S=D in the 8th market (money)."

Now what do you mean by "... the 8th market (money)."??

How do we know that (say) shells are used as money? Because we see one market where apples are traded for shells, a second market where bananas are traded for shells, a third market where carrots are traded for shells...and a seventh market where grapes are traded for shells. **There is no 8th market***. The term "money market" is an oxymoron(?). ALL 7 markets are money markets.

[Seamus Hogab]

But Walras Law is not about what is traded for what. Given a set if prices for each of the seven goods and a pice if money (which can be 1 but doesn't have to be), there will be an excess demand for each of the seven goods and a demand for holdings of money. If S=D in 7 cases, the desired holding of money will equal the stock.

[Nick Rowe]

Seamus: "If S=D in 7 cases, the desired holding of money will equal the stock."

True. But what is a "case"? If S=D in 6 out of 7 *markets*, will S=D in the 7th? Nope. And that was my question.

"...there will be an excess demand for each of the seven goods and a demand for holdings of money." How many excess demands for money will there be? One? I say 7. One for each of the 7 markets in which money is traded. (And, we can't add those 7 excess demands for money up, to get anything sensible at all. See my 9.02am comment above.)

[Roger Sparks]

Recapping, we have 8 markets, one of which is money. Market 7 is grapes which are fixed in price by government action. 6 markets are free to adjust, which results in a price (in money) which clears the market of these 6 commodities.

("Clearing the market" would mean that a new owner is found for 6 commodities.)

Market 7 (the price-fixed market) has become "money" in the sense that it trades at a fixed price against money. A raisin maker (raisins are made from grapes) will know exactly how much money to bring to acquire the grapes he needs but he would not know if he or a competitor would actually get grapes. The raisin maker would not be likely to buy other commodities if he were unable to get grapes.

If we are to make any sense at all out of these 8 markets, they must be bounded by some perimeter. There must be some sense of limits to prevent chaotic blow-up.

A suggestion of using Walras' Law was originally postulated. I think this would suggest a limit on the amount of money available to be used within the 8 market system, but there is no prediction of who went home with all the money, or all the grapes or all the other 6 commodities.

It seems to me like there are a great many combinations that might flow from this model. The maximum amount of money would transfer if the money market was completely exchanged into the hands of the other 7 markets.

The minimum about of money used would result from money never leaving the hands of the money market manager. The other 7 markets could just trade between themselves either in barter fashion or in units based on money but without money as part of the exchange.

Now if we have bounds, there should be no "excess" supply or "excess" demand. The market is the market, demand and supply and price are strictly judgement values in the eyes of the beholder.

Now I agree that the raisin maker who went home with money but no grapes could be said to have "excess demand". I think we all agree that government limits cause market distortion, but the market is the market. The excess demand will just have to await the next market clearing (as will the excess supply of money).

[Roger Sparks]

Typo! In the third to last paragraph, please change "The minimum about of money........." to "The minimum amount of money......." Sorry about that!

[Nick Rowe]

Roger: "Recapping, we have 8 markets, one of which is money."

NO. That is not recapping at all. That is completely wrong.

"Market 7 (the price-fixed market) has become "money" in the sense that it trades at a fixed price against money."

NO. Nonsense.

"If we are to make any sense at all out of these 8 markets, they must be bounded by some perimeter. There must be some sense of limits to prevent chaotic blow-up."

What???

Return to the beginning.

[Nick Rowe]

One last time:

If there are 8 goods, one of which is used as money (the medium of exchange), then there are 7 markets.

That is 7. Not 8. And 9 is right out.

In each of those 7 markets, one of those 7 non-money goods is traded for money.

Money is traded in all 7 markets. That is what "money" means. It means "you can buy and sell everything else for it".

Count. Use your fingers if necessary.

The words "the money market" are an oxymoron. ALL markets are markets for money.

When finance guys talk about "the money market" what they really mean is "the market in which short-term non-money IOUs are traded for money".

Finance guys should not talk like that. But they can't help it; they are only finance guys.

MONEY GUYS SHOULD NEVER TALK ABOUT "THE MONEY MARKET"!

This is the VERY FIRST THING that anyone who studies money should learn.

[Nick Edmonds]

When you're thinking about markets, you don't always want the same level of aggregation. So sometimes you want to think about the market for apples, but sometimes you want to think about the market for fruit, or goods generally, or maybe for granny smiths only.

Likewise with financial assets including money. Sometimes you might want to be more granular and think about balances with Bank A and balances with Bank B. Close substitutes maybe, but definitely distinct claims with different demand and supply.

With n markets, where a market is just in one good or financial claim but not both, this is easy to deal with. But I can't work out how you do it with n-1 markets, where each market is in two items. If the apple purchaser banks with Bank A and the apple vendor banks with Bank B, what other thing is the apple market a market in? Or is it that there is more than one market involved in the transaction?

I'm just interested. Maybe it's something you've dealt with elsewhere.

I don't think it is a fact that there is either n markets or n-1 markets. The reality is there is just a collection of transactions. Grouping these into markets is just a useful way of describing what happens. Sometimes, the n markets way is going to be more useful and sometimes the n-1 markets way will be more useful.

[Roger Sparks]

Nick: You made yourself clear here. We obviously conceptually disagree, but thanks for the comments.

I think I am correct in saying that you perceive money as being a "good" but deny that a market for money exists.

One thing I would like to incorporate into my thinking is that the amount-of-money-available seems to make a difference in the pricing of all other markets. The amount-of-money-available in an economy (or market) obviously changes so, to my way of thinking, we should allow money to have a market with a supply chain of some sort.

Money seems to be created by governments, banks, and counterfeiters. The good produced is the same but method of production different. Following production, there is a path of money movement into the economy (or the market). The initial valuation of the new money is made by the initial taker.

You can see that I think every good (including money) has a supplier.

Next I see the market as an exchange between two suppliers. There is no need for the two suppliers to be also the initial producers of each supply, they only need to have a good to trade.

Finally, I can see that if we do not allow money to have a market, we can define every other possible item to exchange as having it's own market, which results in the number of markets as the number of goods less one. Then, if you have money but seek apples, you go to the apple market and look to make an exchange. But, if you have apples but seek money, you go to the ..money market....?

[Majromax]

@Nick Edmonds:
> With n markets, where a market is just in one good or financial claim but not both, this is easy to deal with. But I can't work out how you do it with n-1 markets, where each market is in two items. If the apple purchaser banks with Bank A and the apple vendor banks with Bank B, what other thing is the apple market a market in? Or is it that there is more than one market involved in the transaction?

I think there's a natural level of aggregation here, where we replace "market" with "auction" (for flexible-price goods) or "lottery" (for fixed-price goods).

At t=0, those who are interested in exchanging goods for money (or vice versa) line up at the auction or lottery. At t=epsilon, the auction or lottery winds up and exchanges are made.

We get different results if markets for all goods open and close simultaneously or if (as contemplated earlier) some markets are open after others have finished.

For your hypothetical, we consider Bank A and Bank B to be equivalent, because money transfers between accounts at each frictionlessly and at par. If we were instead operating in a private money system where banknotes were *not* equivalent, then we'd be breaking the assumption of a singular "money" good.

However, Nick's post here (and Nick's Law) is about going from the 0-money situation to the 1-money situation, not the many-money situation.

@Roger Sparks:
> Market 7 (the price-fixed market) has become "money" in the sense that it trades at a fixed price against money.

With my above in mind, that's where grapes are *not* money. Trading at a fixed price is a necessary but not sufficient condition, because to be money grapes must also be freely convertible, in both directions.

We don't even have two separate monies, because apple and banana vendors are not (by assumption) going to trade directly for grapes.

[Nick Rowe]

Nick E: If there are two goods that are used as money, so you can trade apples for either gold or silver, then it gets more complicated. And there is also a market in which gold and silver are exchanged for each other (which is where we can legitimately talk about "the money market").

But I want to keep it simple here, because this is already confusing enough. Assume there is just one good used as money. (Yep, I am ducking your question.)

"Then, if you have money but seek apples, you go to the apple market and look to make an exchange. But, if you have apples but seek money, you go to the ..money market....?"

NO! You go to the apple market, of course! The apple market is the place where two sorts of people meet: those who have money and want to buy apples; those who have apples and want to use those apples to buy money. The first group creates the demand curve for apples; the second group creates the supply curve for apples.

What we call "the apple market" is, strictly, "the apple/money market", because those are the two goods traded in that market. The only reason we don't call it "the apple/money market" is just because we take it for granted that every other good is bought and sold for money.

[Nick Edmonds]

Majromax - Thanks for your reply, but I was thinking specifically of those situations where we might not want to consider Bank A and Bank B to be equivalent, which might be the case if we were considering the operations of banks for example.

Nick - no worries for ducking my question, but I would point out that it is even more complicated than you suggest, because I wasn't asking simply about what happens when you have two different monies like gold and silver. I was asking how you analyse it where the purchaser pays one type of money but the vendor receives another.

[Seamus Hogan]

Nick. I agree that in your economy, with your definition of "market", if S=D in 6 of the 7 markets, there is no necessity that S=D in the 7th. The statement I can't agree with is "Walras' Law is true and useful for the economy as a whole only if there is only one market in the whole economy, where all goods are traded for all goods. But that is not a monetary economy. And it is not a real world economy."

Walras' Law doesn't refer to markets per se, but to excess demands for goods. The useful thing about Walras' Law is the statement that if there are m independent relative prices, there are m independent S=D condition. Now this is usually defined in terms of some n (where m=n-1), and maybe n is given the name "market", but that is not the point. What matters is that equilibrium can potentially exist as the number of equilibrium conditions equals the number of endogenous variables. That is true in ADM GE, it is true in IS/LM, and it is true in your monetary economy.