Walras' Law says that a general glut (excess supply) of newly-produced goods (and services) has to be matched by an excess demand for some other good. But it could be matched by an excess demand of anything that is not a newly-produced good. It could be an excess demand for money. Or it could be an excess demand for: bonds; land; old masters; used furniture; unobtainium; whatever.
Daniel Kuehn calls this the "Whack-a-mole" theory of general gluts. The excess demand that matches the excess supply of newly-produced goods could pop up anywhere.
Monetary Disequilibrium Theory says that a general glut of newly-produced goods can only be matched by an excess demand for money. There's only one mole to whack. Money is special. A general glut is always and everywhere a monetary phenomenon.
In a monetary exchange economy, Walras' Law is wrong; Monetary Disequilibrium Theory is right. We live in a monetary exchange economy. Walras' Law is the worst fallacy currently taught in economics as gospel truth.
Why is Walras' Law wrong? Why is money special? That's what this post is about.
Keynes failed to escape Walras' Law. He said (somewhere) that if Silvio Gesell succeeded in eliminating an excess demand for money by taxing money, people would switch to wanting to hold land instead, and the general glut could continue. The excess demand just pops up somewhere else. Whack-a-mole. But Keynes was wrong; Gesell was right.
New Keynesian (Neo-Wicksellian) models say that a general glut of goods-produced-today is matched by an excess demand for goods-produced-tomorrow, not by an excess demand for money. That's another application of Walras' Law.
Walras' Law is not a zombie theory. It's a poltergeist. It doesn't realise it's dead (Clower killed it decades ago). You have to first understand why it was once alive before you can persuade it to rest in peace.
So first let's recognise the truth in Walras' Law. Instead of an economy with many markets, where only two goods are traded in each market, suppose that all goods are traded simultaneously in one big centralised market. You take apples bananas and carrots to the market, and exchange them directly for dates eggs and figs, in one big exchange. You give your apples bananas and carrots to the auctioneer, and he gives you dates eggs and figs in return. And suppose further that agents never imagine they might be unable to sell or buy as much as they want to in that market. Because there's no "false trading" at disequilibrium prices — no trade is allowed to take place until the Walrasian auctioneer has found the market-clearing price vector. That's the world of Walrasian general equilibrium theory.
In that world, Walras' Law would be true. Each agent would make one big decision about how much of each good to buy or sell, and that decision would respect the agent's budget constraint. The market value of the goods he planned to buy would equal the value of the goods he planned to sell. Unless agents had fat fingers on their calculators, Walras' Law would be true at the level of each individual agent's planned demands and supplies. Summing across all agents' demands and supplies, Walras' Law would necessarily be true at the aggregate level as well.
We do not live in a world like that.
First, we do have false trading, and agents aren't stupid. They know that if there is an excess supply of apples some of them will be unable to find buyers and so won't be able to sell as many apples as they want to sell. And if there is an excess demand for apples some of them will be unable to find sellers and so won't be able to buy as many apples as they want to buy. They are quantity-constrained. So they revise their planned demands and supplies of other goods taking that constraint into account. The demands and supplies we actually observe in markets are those constrained demands and supplies, not the notional demands and supplies of agents who are stupid enough to think they can buy or sell as much of everything as they wish even at a disequilibrium price vector.
Second, we live in a monetary exchange economy. If there are n goods, including one called "money", we do not have one big market where all n goods are traded with n excess demands whose values must sum to zero. We might call that good "money", but it wouldn't be money. It might be the medium of account, with a price set at one; but it is not the medium of exchange. All goods are means of payment in a world where all goods can be traded against all goods in one big centralised market. You can pay for anything with anything. In a monetary exchange economy, with n goods including money, there are n-1 markets. In each of those markets, there are two goods traded. Money is traded against one of the non-money goods. Each market has two excess demands. The value of the excess demand (supply) for the non-money good must equal the excess supply (demand) for money in that market. That's true for each individual (assuming no fat fingers) and must be true when we sum across individuals in a particular market. Summing across all n-1 markets, the sum of the values of the n-1 excess supplies of the non-money goods must equal the sum of the n-1 excess demands for money.
Walras' Law describes an economy with one market with n goods traded and n excess demands. In a monetary exchange economy there are n-1 markets with 2 goods traded and 2(n-1) excess demands.
OK. So can't we just re-state Walras' Law as saying that the sum of the values of the excess supplies (demands) for the n-1 non-money goods must equal the sum of the n-1 excess demands (supplies) for money?
The short answer is: "No, you can't". Or rather: "You can if you like, but it's a very different beast from the original Walras' Law, and is totally useless". Let me explain why.
Let there be three newly-produced goods: haircuts; manicures; and massages. Let there be one non-money non-produced good: "land". And let there be one medium of exchange: "money". There are four markets: for haircuts; for manicures; for massages; and for land. Money is traded in each of those four markets.
Start in equilibrium. Demand equals supply in each of the four markets. Hold all four prices fixed. Then all of a sudden: the hairdresser decides she wants to buy land instead of a manicure; the manicurist decides she wants to buy land instead of a massage; and the masseuse decides she wants to buy land instead of a haircut. (Substitute "bonds" for "land" and this is the New Keynesian theory of general gluts.)
What does Walras' Law say? Walras' Law says there is an excess demand for land and an offsetting excess supply of newly-produced goods: haircuts, manicures, and massages. A general glut of newly-produced goods, which will cause all three women to be unemployed, because nobody will buy the services they want to produce. WRONG!
Initially, in the market for land there is an excess demand for land matched by an excess supply of money. All three women are trying to buy land with money. And in the markets for newly-produced goods there is an excess supply of haircuts, manicures and massages matched by an excess demand for money. There is one excess supply of money that equals (let's suppose) the three excess demands for money. But what happens next?
Either the price of land adjusts or it doesn't.
If it adjusts, so that all three women stop wanting to buy land at the new higher price, we are back in equilibrium.
If it doesn't adjust, all three women will learn that they cannot buy land, because they can't find a willing seller. They are quantity-constrained in the land market. And being rational utility-maximising agents, they take that new constraint into account and re-formulate their demands and supplies for other goods. What happens next depends on the exact shape of their utility functions. One plausible scenario is that, since they can't buy the land they want, they go back to doing whatever it is they were doing before they suddenly decided they wanted more land. They go back to buying haircuts, manicures, and massages.
Assume my "plausible scenario" is correct. What are the excess demands and supplies we would observe? The hairdresser still drops by the realtor's office, on her way to get a manicure, and asks if they have any land for sale, just in case things have changed. There's still an excess demand for land. She still wants to buy land, even though she might get discouraged from trying to buy some. But then she continues on to buy a manicure. Same for the other two women.
So there's an observed excess demand for land, but no observed excess supply of haircuts, manicures, or massages. Walras' Law fails.
Or does it? Well, you could say that the excess demand for land in the land market is matched by an excess supply of money in the same land market. That's true, totally trivial, and totally useless in explaining general gluts. It tells us what is happening in one market — an excess demand (supply) of the good must be matched by an equal excess supply (demand) for money in that same market — but that tells us absolutely nothing about what is happening across markets.
The only true version of Walras' Law in a monetary exchange economy — the sum of the values of the excess supplies (demands) of the n-1 non-money goods must equal the sum of the n-1 excess demands (supplies) for money — is totally useless because it imposes no cross-market restrictions on excess demands and supplies. It's just the trivial sum of the individual market restrictions. And imposing cross-market restrictions on excess demands and supplies is precisely what Walras' Law was supposed to do.
Let a be the excess supply of apples, and let A be the excess demand for money in the apple market. Let b be bananas, and c be carrots. If a=A, and b=B, and c=C, then whoopee! a+b+c=A+B+C. But that tells us nothing about the relation between a,b,and c. On the other hand, a+b=C would be useful, but unfortunately it's also false.
Walras' Law is either useful or true. Pick one.
Back to my three women. Start back in equilibrium. Hold prices fixed. Then all of a sudden all three women decide they want to hold more money and less land. Walras' Law says that there's an excess supply of land matched by an excess demand for money. But no excess supply of newly-produced goods like haircuts, manicures, or massages. WRONG!
Initially the three women do plan to sell land and hold the proceeds as money. There's an excess supply of land and an excess demand for money in the land market. But what happens next?
Either the price of land adjusts or it doesn't.
If it adjusts, so that all three women stop wanting to sell land at the new lower price, we are back in equilibrium.
If it doesn't adjust, all three women will learn that they cannot sell land, because they can't find a willing buyer. They are quantity-constrained in the land market. And being rational utility-maximising agents, they take that new constraint into account and re-formulate their demands and supplies for other goods. What happens next depends on the exact shape of their utility functions. One plausible scenario is that, since they can't get extra money by selling land, they try to get extra money by selling more haircuts, manicures, and massages. But that won't work either, since they can't find extra willing buyers.
A second plausible scenario is they they try to get more money by buying less haircuts, manicures, and massages. That's what causes an excess supply of newly-produced goods. That's what causes a general glut.
Nobody can ever stop you buying less other goods.
What makes money different? What makes money special?
In Walrasian General Equilibrium theory, there is one big market in which all n goods are traded. For each good, and for each person, that person is either a buyer of that good or a seller of that good — not both. If you are a buyer and the good is in excess demand, you won't be able to buy more. If you are a seller, and the good is in excess supply, you won't be able to sell more. You can't always do what you want — you need to find someone else willing to take the other side of the trade.
Money is different because it is the only good for which every person is both a buyer and a seller. Money comes in one hand when you sell all other goods, and goes out the other hand when you buy all other goods. Each person is only on one side of the market for every other good. Every person is on both sides of the money market. That's because there is no "money market". Or, rather, every market is a money market in a monetary exchange economy. So it doesn't matter if markets are in excess demand or excess supply. The individual can always get more money by buying less other goods even if he can't get more money by selling more other goods.
But while the individual agent can always get more money by buying less other goods, that's not necessarily true in aggregate. And the individuals' attempts to do something which is collectively impossible are what cause the general glut of other goods.
Leave money out of the picture, and it becomes impossible to explain why there should be a general glut of haircuts, manicures, and massages. If all three women are unemployed, why don't they just do a 3-way barter deal? The hairdresser gets a manicure, the manicurist gets a massage, and the masseuse gets a haircut. All three women are better off. Any theory of general gluts which ignores monetary exchange and the excess demand for money assumes the three women are irrational in leaving unexploited gains from trade lying on the table.
Worthwhile Canadian Initiative 
Paine: “All this isn’t shaving the obvious
Your three service toononomy with
Land and money
U count 4 markets
What no market in money
Of course not that’s credit
And that’s not possible”
A market in “credit” is really just a market in bonds. You give me money; and I give you an IOU. That IOU is a bond. And my “land market” could easily be replaced by a “bond market” without changing the rest of the story.
david: “The point being that it is not problematic to find money being in excess constrained demand on one market and in excess constrained supply on another market – that’s only because we’ve constrained it, by obliging all agents to spend all their endowment. This prohibits general gluts immediately, of course.”
Agreed. We can also imagine that haircuts are in excess supply while massages are in excess demand. And if the excess demands and supplies roughly balanced, we wouldn’t talk about a general glut or aggregate-demand-deficient unemployment. We would talk about structural unemployment.
Andy: I disagree. In an old post I sketched a world where bling was the medium of account, and there was a demand for bling that looked exactly like a standard money demand function. We get utility from wearing bling, but it’s the real value of bling that we get utility from (because we want to show off our wealth), and the (stock) demand for bling increases with income and decreases with the opportunity cost (interest rate). But it’s a barter economy. If prices are sticky in terms of bling, you can get an excess demand for bling, but it doesn’t affect the rest of the economy.
david stinson: if you could not hoard money (if the government passed a law saying that all money had to be spent in less than 1 month, so velocity is fixed at 12) then it would be hard to get recessions. (Except people would get around the law).
JP: I think that’s a lot of it. But it’s hard to disentangle differences of using language from different ways of viewing the world.
Nick (re: your conversation with Andy),
I’ve read your money-as-bling posts (if only Kocherlakota were that creative he’d have called it “Money for Noting” instead of “Money as Memory” and have groupies), and your rent control analogies, but I still don’t understand how you think you can isolate the importance of monetary MOE function as causing these disequilibrium recessions. It seems like both the MOA and MOE roles are inextricably necessary.
If the rate between the MOA and the MOE was not fixed, then changes in demand for the MOE would not cause recessions because it would merely result in a change in the price of the MOE (defined as the MOE/MOA exchange rate) but not the price level (so there would be no reason for stickiness). Prices in the store (quoted in MOA, with the MOA/MOE exchange rate presumably posted) would be unchanged but real MOE balances would change to meet demand. Prices in terms of the MOE would then be presumed flexible since it could change in one market with no change in the price level.
Changes in the demand for the MOA might cause a NK recession (relative price problems, not quite like rent control) as rigid prices tried to move clunkily, but wouldn’t cause a monetary disequilibrium recession because people couldn’t freeze by refusing to buy anything and instead holding the MOA (because it is not the MOE). This seems true even if barter weren’t an option. If prices refused to move upon an excess demand for the MOA, then you’d just have the next best behavior as people gave up on their excess demand for the MOA (like rent control). The bottom line is that they couldn’t accumulate the MOA by refusing to spend, so they would surely give up sooner than later.
Only if the MOE=MOA or the MOE/MOA rate is fixed (making them equivalent for this purpose) can you have the unique situation where the price of the very thing that people can decide to hold to avoid spending be expected to fail to clear.
Your money-as-bling post posits money as the MOA (with bling creating a real demand function) with no MOE (barter), merely showing that the MOA can’t do the job by itself, but seems to ignore the fact that the MOE also can’t do the job by itself (create a disequilibrium recession). It isn’t that MOE is more important than the MOA when it comes to causing these kinds of recessions, it’s just that barter by definition happens to eliminate an MOE (but not necessarily MOA). But that doesn’t make the MOE function any more important than the MOA function. I think you need “money” to be both the MOE and the MOA — including by fixing the MOE/MOA rate — to create general gluts. I also don’t think it is useful to talk about which of these functions has the “real power” as Andy does, except from a NK perspective where it seems to be plainly the MOA function. But if you believe in the freezing, disequilibrium recession, I can’t imagine a helpful answer to which function is more important or which cause more proximate.
Let’s me honest. A stake had already been put through “Walras’ Law” by Wicksell & Hayek, decades before Clower. And by all sorts of “classical” economists before them.
Nick writes,
“Walras’ Law is not a zombie theory. It’s a poltergeist. It doesn’t realise it’s dead (Clower killed it decades ago).”
Side note: Hayek’s argument in TPTofC is that “Say’s Law” can be nothing other than the same equilibrium condition specified by “Walras’ Law”, i.e. an equilibrium condition for a construct which does NOT include money, credit, finance, leverage, and banks …
dlr: Hmmm. A thoughtful well-reasoned comment. I was already agreeing with you until the last paragraph. But I now think you are right on that last paragraph too.
Greg: my understanding rests on Clower. I don’t remember reading about Walras’ Law in Hayek or Wicksell. But that might just be my memory, or my failure to read widely and thoroughly enough.
Nick, re: Bling.
The requirement is that there be a medium of exchange, in order to have recessions, but once the medium of exchange exists, it has no power. As I said in my analogy, the figurehead may be necessary for the maintenance of civil order, but the figurehead has no power. Merovingian France couldn’t operate without a monarch, but the particular actions of the monarch were irrelevant to the particular functioning of the kingdom. (Typically, also, the ultimate result of separating the MOE from the MOA would be what happened in France: the old MOE is discarded and replace by the Carolingian MOA.)
I don’t think I had quite thought things through first time. I hope readers don’t mind a second attempt.
Start by considering a simple economy with n goods. There is an auctioneer who only allows trade at market-clearing prices, so the economy is Walrasian. There is a numeraire, and possibly some commodities (eg gold) with money-like characteristics, but no fiat money. The auctioneer starts the process of price discovery by announcing a randomly-chosen vector of prices, and agents respond by revealing their plans to buy and sell at those prices. Since each agent plans to stay on his or her budget constraint, for each individual Sum(i)pEi(p) (the sum of planned transactions Ei(p) at the announced prices) will be zero, regardless of whether these prices clear markets. (I use Sum(i) to mean summing over commodities). Summing over all agents (denoted by Sum(h)) gives Walras’ Law – the double summation Sum(h)Sum(i) pEi(p) = 0. This implies that if we define E(p) as market excess demand (a sum over agents), the sum over all commodities Sum(i) pE(p) = 0. Since this is an economy without money, Walras’ Law and Say’s Law (in one interpretation of Say) are the same proposition: excess supply of all commodities is ruled out. Eventually the auction process converges to a set of market-clearing prices p*. At this point all n components of pE(p) are zero.
Now suppose that the auctioneer gives up and goes home before p* has been reached, and individuals start to trade at a set of disequilibrium prices p+. They still plan to stay on their budget constraint, so notional market excess demands (NE(p)) will still obey Walras Law – Sum(i) p+NE(p+) = 0. But since markets don’t clear at p+ not all trades will get made, and there will need to be a rationing rule: those who cannot sell their massages will need to cut back on consumption of manicures, and so on. Ultimately (since for all trades which do take place there is both a buyer and seller) it will be true that in each market observed excess demand (OE(p+)) will be zero, so we have a relationship which looks just like Walras’ Law, but is actually quite different – the sum, Sum(i) P+OE(p+) is still zero, but individual components are quite different. In addition, each individual is still satisfying a budget constraint: but for an unemployed individual this budget constraint is different from, and ‘tighter’ than, the notional budget constraint to which he was conforming during the auction.
At this point there are two distinct approaches to the problem. One approach argues that since a general glut should be defined as excess supply (eg E(p) < 0) for all commodities, it can only occur in a monetary economy: in this case (by Walras’ Law) it corresponds to an excess demand for money. The alternative, which is what I was trying to get across, says that we have to break down the excess demand terms into demand and supply components (say (D(p) and S(p)). Then notional demands are ND(p), observed demands OD(p). It is quite possible that, for every commodity, OD(p+) is smaller than ND(p+) – all individuals would like to sell more at the current set of (disequilibrium) prices, but cannot actually do so because of coordination failure. It’s also possible that OD(p+) is smaller than ND(p*): everyone would willingly trade more if the auctioneer did his job properly rather than leaving early.
So far there’s no money. In my view the important thing about money (in addition to its role as medium of exchange) is not that it provides the extra commodity which would permit disequilibrium in all goods markets. Instead, it is that it is the only commodity where individuals know that(ignoring hyperinflation) they will not face rationing. I can ‘sell’ as much money as I want in return for bread, whereas I may not be able to sell unlimited quantities of my labour if there is unemployment, or bonds if there is a financial crisis. So holding money means that it is less likely that I will be forced off my notional demand curve for consumption goods: and the easier it is for individuals to realise their notional demands, the less the risk of a general glut (or Keynesian unemployment equilibrium). In ‘normal’ times financial intermediation means that non-money financial instruments (‘credit’) can play a similar role for an individual, but this will not work if everyone tries to draw on them simultaneously (when there will be a financial crisis). In a crisis only money will do, and will appear to be in excess demand, with excess supply everywhere else. But if the auctioneer did his job properly the crisis would not occur.
“Walras’ Law” is just an equilibrium condition of a math construct — the central ideas of the construt predated 1960s math economics.
Nick writes,
“Greg: my understanding rests on Clower. I don’t remember reading about Walras’ Law in Hayek or Wicksell. But that might just be my memory, or my failure to read widely and thoroughly enough.”