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.

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.

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.

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.