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“If prices are not at market-clearing levels, then people on the long side of a market will be quantity-constrained. They will be unable to trade as much as they want to trade. A quantity constraint in one market will affect demands in other markets.”

It’s relatively easy to prove this is possible in a monetary economy.

But, to be clear, you are saying that this cannot happen in a barter market?

I.e., you are saying it is impossible for “people on the long side” to end up with any unsold output in a barter economy?

Are you sure?

Why is it simply not possible for the market clearing “price” for a particular good in a barter economy to be zero?

Why can’t zero be one of the equation coefficients?

“You can't sell more than people want to buy.”

Why is it not possible in a barter economy for a particular good to be absolutely unwanted? Why is it not possible for that single auctioneer calculation to leave somebody out in the cold? If so, this will certainly be a catalyst for a recession.

(7:00 a.m. is indeed better than 3 a.m. for efficient intellectual clearing. But it is probably a bit early for optimization.)

Here's something I have no business asking for: I would very much like to read an expanded version of this post, with some more references, and more hand-holding over the text book economics you assume we already know. (If there was some way of me paying you for it ...) As I recall, excess demand was only ever mentioned by way of explaining what equilibrium looks like in an Edgeworth Box, in first-year, I think - as far as I can recall, I've never had to think about these things before. I don't think I came across a demonstration ending "therefore the sum of excess demands must be zero". In 2nd year we were too busy with signaling, contract theory, game theory, and such like. I'm a recent graduate student, we didn't cover this in graduate courses either. Are these the dark ages?

This makes me think of the chapters we never reached in Mas-Collel. I don't know if you're familiar with that book, but would you say it "gets it wrong" too, because it stays in the world of a Walrasian auctioneer?

fourthtimeanon: "I.e., you are saying it is impossible for “people on the long side” to end up with any unsold output in a barter economy? Are you sure?"

No, I'm not at all sure! In fact, I'm pretty sure you can indeed be on the long side in a barter economy. I can't have been clear enough. One of the things I got a bit muddled on in my previous post was the distinction between a barter economy with n(n-1)/2 pairwise markets, with 2 goods trading in each, and an economy with a centralised auctioneer, with one big market where n goods trade. Both are non-monetary economies, but that doesn't mean they are the same. I'm still trying to get my head round whether they end up with the same equilibrium set of quantities traded, for a given price vector.

I didn't think about the cases where a good might have a zero price in equilibrium. Don't think it matters, since if p=0 the value of the excess demand must be zero too.

anonymous: Every age is a Dark Age in some respects. We can't remember everything. Indeed we progress by finding new better simpler ways of doing things, so we can forget some of the old ways. But sometimes we find we need stuff we have forgotten after all.

Does anyone know of any simple standard derivation of Walras' Law on the web?

I haven't heard of Mas-Collel. But then I'm out of touch anyway. Most (all?) micro texts stay in the world of the Walrasian auctioneer, so they are wrong whenever trade takes place at non-market clearing prices. In other words, they don't work outside of full market clearing equilibrium.

The short version of your complaint is that the standard economist's starting point is: Assume perfectly liquid markets.

Or alternatively: the concept of competitive equilibrium is fundamentally illogical because every individual is required to act "as if" there is infinite supply and demand at the given price even while knowing that supply and demand are finite and will in fact be the determinants of the price.

"Does anyone know of any simple standard derivation of Walras' Law on the web?"

http://www.econ.hku.hk/~wsuen/micro/ge.html

Thanks Patrick. Exactly what I had in mind.

Apostate: Even if markets are perfectly liquid, and perfectly competitive, someone won't be able to buy or sell as much as they want if the price is wrong.

Varian, Kreps and Mas-Collel are the current, classical big 3 of graduate-level micro. Varian is traditional, and accessible and geometric; Kreps is game theory; Mas-Collel is too much math.

Hmm. I thought the definition of a perfectly liquid market was that you could sell or buy as much as you want without affecting prices.

There must be a reason the most important terms in economics have many definitions.

Apostate: "the concept of competitive equilibrium is fundamentally illogical"

A better way of stating the conditions of competitive equilibrium is to say that everyone behaves non-strategically. Obviously, no one expects to observe cases of completely non-strategic economic behavior, but one, nevertheless, wants to work out the logic of markets, before introducing the complicating factor of strategic behavior.

fourthtimeanon: "Why is it simply not possible for the market clearing “price” for a particular good in a barter economy to be zero?"

NR: " an economy with a centralised auctioneer, with one big market where n goods trade. Both are non-monetary economies"

Hey, fellas, let's play by the rules, now. There are no "prices" in a barter economy. Prices only make sense in a monetary economy. Money has at least two functions: Number 1 is to enable enumeration; store-of-value-as-claims is number 2. Money is the language of prices. Without money as an enumerator, the Walrasian auctioneer is rendered mute. Lacking money, the language of prices, a centralised auctioneer cannot perform his function.

In a barter economy, every seller is her own auctioneer, able to calculate in terms only of what she is selling, which good is common to her purchase transactions. She's going to trade away the good she's selling until it has some zero or positive marginal value to herself.

I suppose there might be goods, which are really bads: someone could be selling trash disposal. That little complication is the answer to whether there can be genuine non-clearing. If the good has zero-positive marginal value to the seller, that's the market-equilibrium, and a perfectly sensible one: shouldn't that be the natural outcome of endowment-unconstrained barter transactions? If the "good" is really a bad, the seller will "pay" to dispose of it, until self-disposal makes sense, until the seller has reached a zero-negative marginal value transacting in markets for disposal.

This is not the same as carrying an inventory. In a barter economy, an inventory implies incomplete markets, I think -- you've chosen not to have time-marked transactions: you don't have the possibility of agreeing to deliver x lemons for y apples at time t. Your seller of lemons chooses to hold an inventory of lemons against the expectation of engaging in future transactions in markets yet to come into existence, with greater expected present value than available present and possible exchanges.

BW: I object strongly to the term "better" in your response. I can agree that as a field of study economics is difficult enough that studying non-strategic behavior and competitive equilibrium has value. But I also believe that its (limited) value would be better appreciated if we could all agree that it is fundamentally illogical to let demand and supply determine prices while at the same time asking members of the economy to act as if they have no knowledge of demand or supply.

Nick:

Please stop wasting time with barter.

Stick to the real world--monetary economy without an auctioneer. (Yes, I understand how you get into barter, though I think accusing Cochrane of assuming a Walrasian Auctioneer rather than barter is more plausible, right?)

n-1 excess demands for nonmoneary goods. And then, n-1 excess demands for money. So, 2(n-1) excess demands. Don't we usually sum up the excess demands for money to get one grand total? "The" excess demand for money adds the excess demand for money that matches the surplus of shoes and the excess supply of money that matches the shortage of apples, and so on. Doesn't the sum of these individual market excess demands for money add up to offset the net excess demands for all the other markets?

Yeager's complaint about Clower (as best I understood it) was that if "max U" with capacity contraints reduces real income, that will reduce the demand for money. This will lead to an excess supply of money and raise expenditure on something. Where is that macro impact in Clower's approach?

Suppose we have monetary equilibrium. The quantity of money is sufficient to meet the demand to hold money at the current level of nominal income. Money demand is negatively related to nominal income. If shortages of somethings and surpluses of other things result in quantity constrained utiliity maximization problems, so that real output falls at given prices, then real income falls, money demand falls, and we have an excess supply of money. That money is spent on something, raising demand, output, and income back again.

If we are in this situation, there is nothing that requires that every market clear. But if nominal income (and total expenditure) matches productive capicity, it seems to me that shortages and surpluses must match. To me, this is what monetary equilibrium means--the shortages and surpluses match.

Now, if the quantity of money is too low, and output falls so that the demand for money falls to meet the given supply, then there is no "effective" excess demadn for money given the lower real income. It is notional. If real income was at capacity, then there would be an excess demand for money. The current quantity of money is too low relative to the demand when real income is at capacity.

It seems to me (and I thought to Yeager) that the Clower business requires that the quantity of money passively adjust to changes in the demand for money causes by changes in output because quantity depends on the short side of the market.

It may all be fine as a formal explanation of what Keyens was trying to say. But it requires a liquity trap or else an income elastic supply of money. It is like, if we have a monetary policy based upon the real bills doctrine, gee, disaster might ensure.

If, on the other hand, we change the quantity of money to target a growth path for nominal income..then exactly how is max U subject to prices and quantities contrained by the short ends of the market relevant?

Perhaps you can help explain this to me.

Suddenly, everywhere I look,I see disequilibrium Walrasian arguments. We are rediscovering Clower and Leijonhufvud.

(Actually, these were before my time - I am 36 - so I didn't get any of this in grad school [at U of Chicago] or even in my own outside readings, mostly focused on New Keynesian macro. And I did read a lot outside Lucas, Sargent and Prescott and more modern incrarnations.)

Good piece on this broader topic by David Laidler (2009) and Robert Gordon (2009).

Fundamentally, though, these are stories about coordination failure across markets. Question is: do we think these disequilibrium based coordination failure arguments are what generates the violation of Say's law in reality?

Apostate: "Hmm. I thought the definition of a perfectly liquid market was that you could sell or buy as much as you want without affecting prices." But that also sounds like a definition of (lack of) monopoly or monopsony power! There isn't really an agreed-on definition of liquidity.

Bruce: I generally agree with the other things you say, but totally disagree with this paragrapgh: "Hey, fellas, let's play by the rules, now. There are no "prices" in a barter economy. Prices only make sense in a monetary economy. Money has at least two functions: Number 1 is to enable enumeration; store-of-value-as-claims is number 2. Money is the language of prices. Without money as an enumerator, the Walrasian auctioneer is rendered mute. Lacking money, the language of prices, a centralised auctioneer cannot perform his function."

Prices are just exchange rates between 2 goods; they do exist in barter. The Walrasian auctioneer can use any of the n goods (or even an imaginary good) as a unit of account. What is crucial about money is its role as medium of exchange, and you can only tell which good is the medium of exchange by looking at what markets exist.

Bill: "Please stop wasting time with barter. Stick to the real world--monetary economy without an auctioneer. (Yes, I understand how you get into barter, though I think accusing Cochrane of assuming a Walrasian Auctioneer rather than barter is more plausible, right?)"

Yes, I think you are right. Monetary exchange economy, vs Walrasian Auctioneer.

Next bit is crucial. I'm going to quote Bill at length:
"n-1 excess demands for nonmoneary goods. And then, n-1 excess demands for money. So, 2(n-1) excess demands. Don't we usually sum up the excess demands for money to get one grand total? "The" excess demand for money adds the excess demand for money that matches the surplus of shoes and the excess supply of money that matches the shortage of apples, and so on. Doesn't the sum of these individual market excess demands for money add up to offset the net excess demands for all the other markets?"

My answer to Bill:
1. Yes.

2. BUT, the sum of the n-1 excess demands for money can be a VERY misleading statistic. Take the unemployed worker who is looking for a $1,000 per month job. Excess supply of labour $1,000. Excess demand for money in the labour market $1,000 per month. Zero excess demands elsewhere. It LOOKS LIKE he wants to add $1,000 per month to his stock of money. But of course he wants to spend it , if he ever gets a job.

3. BUT, it's true trivially. Since in each of the n-1 markets the excess demand for money will equal the excess supply of the other good traded in that market, it is trivially true that the sum of the excess demands for money will equal the sum of the excess supplies for the other goods. It is not something that follows from the whole budget constraint. If M1=G1, and M2=G2, and M3=G3, etc., then obviously/trivially the Sum of the M's = the Sum of the G's. In other words, it's almost a definition of a market that people offer to sell an equal value of goods to what they ask to buy. We gain absolutely nothing by adding up this definition across markets.

I'm gonna think about the rest of your comment. Don't think I understand it all yet.

Arin: I'm very lucky (for this topic); I'm 54! But the area was already dying when I was in grad skool.

"Fundamentally, though, these are stories about coordination failure across markets. Question is: do we think these disequilibrium based coordination failure arguments are what generates the violation of Say's law in reality?"

Yes! It does not explain why prices don't adjust instantly, or what caused the initial shock to AD, but if we have (significant) non market clearing across the economy, this is the only way we can understand the consequences.

Arin: You should have said: "Suddenly, everywhere I look,I see disequilibrium [NON-]Walrasian arguments. We are rediscovering Clower and Leijonhufvud."

“The only case where n goods result in exactly n excess demands is with a centralised (Walrasian) auctioneer, who matches demands and supplies for each on the n goods simultaneously. Think of it as a single market but with n goods traded instead of the normal 2 goods traded. That is the only case where it is true that the sum of the n excess demands equals zero. But that is not the real world, anywhere.”

I’m still trying to figure this out.

So are you saying that the only market in which Say’s Law holds is that with a centralized (i.e. simultaneous) auctioneer?

And are you saying Say’s Law does not necessarily hold in a bilaterally functioning barter market?

If that is what you are saying, then I’m a believer. It’s pretty well equivalent to what I’ve been maintaining, given “that is not the real world, anywhere”. In fact, one can argue it’s not really a market at all, but a mechanism for mandatory clearing, which conforms to Say’s Law by construction, after the fact.

"Without money as an enumerator, the Walrasian auctioneer is rendered mute. Lacking money, the language of prices, a centralised auctioneer cannot perform his function."

I think the auctioneer is smarter than that. Just involves juggling matrices a bit.

"Stick to the real world--monetary economy without an auctioneer."

Agree. But you NR introduced the barter case. I'd like to know how you qualify it (central, bilateral, etc.) and whether its true or not.

"Why is it simply not possible for the market clearing “price” for a particular good in a barter economy to be zero?"

My question. I'd still like to understand an answer, given that you agree thank goodness that prices exist in a barter economy. This is the answer that proves gluts are possible in any barter economy.

"BUT, the sum of the n-1 excess demands for money can be a VERY misleading statistic. Take the unemployed worker who is looking for a $1,000 per month job. Excess supply of labour $1,000. Excess demand for money in the labour market $1,000 per month. Zero excess demands elsewhere. It LOOKS LIKE he wants to add $1,000 per month to his stock of money. But of course he wants to spend it , if he ever gets a job."

I think this is wrong. There's a difference between excess demand corresponding to realized output and current clearing requirements for that output, versus projected excess demand based on contingent future output.

Nick:

I must be confusing something.

I follow along (and I was very much influenced by Leijonhuvud, and read Clower, of coursre.)

But then, in reading your response, I see the "supply" of money that is supposedly matching the demands for other things as being income. And income isn't the quantity of money. And the supply of goods isn't the demand for money. The demand for money is the amount of money people want to hold and the "supply" of money is the quantity.

I don't mean to suggest that you don't understand this. And there are obviously stock flow issues. Perhaps I just don't understand, but there seems to be a problem of approarch there.

As long as money is flowing through cash balances, and there is no effort to accumulate them, then income and expenditure must match in aggregate. And so, the shortages and surpluses of particular things offset one another.

If there is an effort to change cash balances, then they won't match until somehow there is no longer an effort to change cash balances.

Bill: whenever I say "excess demand for money" above, I do indeed mean an excess FLOW demand for money. I have to, since the excess supplies of all the other goods are flows.

But I don't see how this creates any particular difficulty. The unemployed worker in my example has an excess flow demand for money of $1,000 per month, matching his excess flow supply of labour. So it LOOKS like he wants to add to his stock of money at the rate of $1,000 per month. But of course he doesn't. He (probably) wants to spend (about) an extra $1,000 per month on goods, and keep his stock of money roughly constant (or increase it a little bit).

In a monetary exchange economy, income IS (mostly) a flow of money. And the supply of goods IS a flow demand for money. But as Laidler always says: "they want to HAVE the money, but that doesn't mean they want to HOLD it". (Never sure if he's punning off wedding vows!)

"As long as money is flowing through cash balances, and there is no effort to accumulate them, then income and [DESIRED] expenditure must match in aggregate." YES, but perhaps at less than full employment. "And so, the shortages and surpluses of particular things offset one another." NO, if we are at less than full employment.

fourthtimeanon: sorry, but you lost me there.

I agree with those who have chimed in that prices do exist in a barter economy, just select one good as a numeraire.

This (interesting) discussion has missed an important aspect: "money" is not a good, it is an asset. (It is amazing how frequently people miss this basic characteristic.) As such, it represents deferred consumption today for consumption in the future. Thus, if you introduce a non-perishable asset into your hypothetical economy, you add the possibility of intertemporal exchange of consumption, and your set of goods (think dated-commodities) gets very, very large (economist usually take a short-cut and call it inifinite). Obviously the market clearing conditions are more complicated under these conditions.

The bottom line: don't treat money as a perishable good, okay?

R.Chun, if I plan to defer my consumption for longer than the time it takes me to shop wouldn't I buy a bond? (At least in the usual case of positive interest rates)

That's a fair question. Clearly "money" has characteristics that bonds do not; for example liquidity, divisibility, and lower risk. Also, if we are talking about our toy Arrow-Debreu economy here, you'd need story about why an agent would offer a bond at a positive interest rate. This gets you into time preferences and the social environment that supports enforcement of claims like bonds. The same issue applies to money too, but it is interesting that stuff functioning as stores-of-value can organically arise without, say, an extensive legal system. I'm thinking of cigarettes in a POW camp and the like. (But again our mathematical environment is silent about the mechanism design.) My comment was motivated by Nick's exogenous introduction of money into the mix.

NR: "Prices are just exchange rates between 2 goods; they do exist in barter. The Walrasian auctioneer can use any of the n goods (or even an imaginary good) as a unit of account."

And, if the auctioneer uses any of the n goods as a unit of account, then that's money.

All I'm saying, is that you might want to be analytically careful about the distinction between money as a unit-of-account and money as a store-of-value or medium-of-exchange.

Let's say you have 5 goods, but no money. Then, there are 10 markets, and 10 ratios of exchange. If the Walrasian auctioneer comes along, and does his job, and achieves a general equilibrium among those 10 markets, he can sum up his results in 4 prices, using one good as numéraire. 4 prices for 10 markets. If.

Bruce: Agreed. (Just to explain Bruce's math: 10=n(n-1)/2 when n=5).

But to me, the key property of money (in this context) is it's use as medium of exchange. That's what I intended to mean by "money" in this post.

R Chun: assets are goods too! And money is not unique in being a store of value; lots of goods fill that role. And cigarettes weren't just a store of value in POW camps; they were the medium of exchange.

Reposted from the prior thread:
Nick:

1) You've got the causality backwards. Says law is that if prices adjust, all markets clear.

2) What I mean is that you can write a series of equations representing the relative value of every good versus every other good. But these equations are not linearly independent. You can always reduce to a basis identical to your monetary economy.

3) I think you have me there. It cannot be the supply curve that matters like that. Let me try again then, it must then simply be the conflict between the unit-of-account (price-stickiness) and store-of-value role of money (cash balances). When people demand to hold more money, implicitly they are raising its value versus all goods. But when people are price sticky, they are also trying to hold the value of each good with respect to money. The result is over-constrained. Perhaps this what you've meant to say all along.

Nick - yes, I meant to have said "NON Walrasian" ...

Didn't Coase address this sort of market failure?

Jon: I think you and I have very different definitions of Say's Law. On your point 2, I would agree that in a Walrasian economy, with n goods, the n equations for excess NOTIONAL demands are linearly dependent. I think of that as Walras Law. But a monetary exchange economy is not a Walrasian economy. And constrained excess demands are not the same as notional excess demands.

Arin: good! That clears that up!

Chris: I think the short answer is "no"; Coase wasn't talking about anything like this. He was definitely thinking micro/partial equilibrium. But I expect you could try thinking about it in Coasian terms. After all, it's transactions costs that explain why people use monetary exchange. If there were no transactions costs, we could imagine millions of unemployed workers with an excess supply of labour getting together with thousands of firms with an excess supply of goods, and all doing one massive big multilateral and multi-good barter exchange to get back to full-employment. The Walrasian auctioneer would do the same thing. One big market for everyone and everything.

Nick writes:


On your point 2, I would agree that in a Walrasian economy, with n goods, the n equations for excess NOTIONAL demands are linearly dependent. I think of that as Walras Law. But a monetary exchange economy is not a Walrasian economy. And constrained excess demands are not the same as notional excess demands.

Nick, I understand that that is your CLAIM, but you haven't offered a substantiation that what you say is so. Given if a then b, one cannot claim ~b simply because a is false.

Now I haven't substantiated my CLAIM either, so my point is limited: if you want to make a compelling case you'll need to explicitly explain why the supports are independent or not.

Jon: agreed. I'm trying to find the right thought-experiment to address this question properly. Let's see if my latest post (rent controls) works.

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