It is money (qua medium of exchange) and only money, that makes Say's Law wrong. And those Keynesians who reject Say's Law but who also reject the uniqueness of money are as wrong as those who accept Say's Law.

Say's Law is wrong, and the fact that it is wrong is really important for macroeconomics. But it's not obvious why Say's Law is wrong. And you have to get right inside the minds of those who think Say's Law is right if you want to change their minds. That's what I'm trying to do here.

For a whole economy, does planned expenditure on newly-produced goods necessarily equal planned income from the sale of newly-produced goods? That's what I mean by "Say's Law" in this context.

Let me start out by "proving" Say's Law.

There are n goods, and each individual has a budget constraint which says his planned purchases of goods must have the same value as his planned sales of goods, summed over all n goods. If we add up all the individuals' budget constraints we see that the value of the excess demands must equal the value of the excess supplies (aka Walras Law).

(So far, things are not looking good for Say's Law, because there could be an excess supply of newly-produced goods matched by an equal excess demand for, say, bonds, if everyone wants to save part of their income rather than spend it on newly-produced goods. But wait a minute.)

What determines the actual quantity traded in a market where there is excess demand or excess supply? The standard assumption is that the short side of the market determines quantity traded, since you can't force people to buy more than they want to, or sell more than they want to. Exchange is voluntary. For goods in excess demand, quantity traded is determined by supply. For goods in excess supply, quantity traded is determined by demand.

With some markets not clearing, people are forced to revise their plans, since for some goods they cannot sell all they want; and for other goods they cannot buy all they want. These revised plans must also be consistent with the individuals' budget constraints, which means a constraint on an individual's purchases or sales for one good must spillover to affect his planned purchases and/or sales of some other goods.

For example, suppose people were initially planning an excess demand of $100 worth of bonds financed by an excess supply of $100 worth of newly-produced goods. But if you realise you can't actually sell an excess supply of $100 worth of newly-produced goods (and you can't, given the short-side rule) you realise you can't finance an excess demand of bonds with the proceeds. Equivalently, if you realise you can't actually buy an excess demand of $100 worth of bonds (and you can't) you realise that selling an excess supply of $100 worth of newly-produced goods would violate your budget constraint.

Therefore there cannot be an excess supply of $100 worth of newly-produced goods. Say's Law is right. In aggregate, people must plan to buy newly-produced goods of a value exactly equal in value to what they expect to sell and earn as income. They would violate their budget constraints otherwise.

(Just translate the above into math, and write "QED" underneath.)

What's wrong with the above "proof" of Say's Law? (Apart from the fact that it "proves" that no market can have an excess demand or supply?)

In a word: money. Money, qua medium of exchange, is weird. Every other good has one market, which is either in excess demand or in excess supply. But money is in every market.

For every other good, an individual is either a buyer, or a seller, but not both. An individual is either on the short side, or the long side, but not both. But for money, every individual is both a buyer and a seller.

Suppose initially there is an excess supply of newly-produced goods matched by an equal excess demand for money. If money were like bonds, people wanting to buy money would be on the long side of the money market, and would be unable to buy more money and so would stop trying to sell more goods. But money is not like bonds. Every individual is on the long side of the money market when he tries to earn income by selling newly-produced goods, but on the short side of the money market when he tries to spend income by buying newly-produced goods. If you can't get more money by selling more newly-produced goods, you can still get more money by buying less newly-produced goods.

(And if newly-produced goods are initially in excess demand, and money in excess supply, you can't get rid of money by buying more goods, but you can still get rid of money by selling less goods).

If there are n goods including money, there are n-1 excess demands or supplies for the n-1 non-money goods, and also n-1 excess demands or supplies for money, giving 2(n-1) excess demands or supplies in total. Any "proof" of Say's Law that counts only n excess demands or supplies must be wrong (except when n=2, but then money would be redundant anyway, if there were only one non-money good). The most we can say is that in each of the n-1 markets, the excess demand or supply of the non-money good in that market must equal the excess supply or demand for money in that market. (And a true barter economy where everything can be traded for anything else has (n-1)n/2 markets, not the n markets of my "proof").

Brad DeLong interprets John Cochrane as implicitly assuming that velocity is fixed. I interpret John Cochrane as implicitly assuming a non-monetary economy. Both interpretations make sense. But I think (hope?) mine is more fruitful. It is not obvious that my "proof" of Say's Law is wrong, and that it is only the weirdness of money that makes Say's Law wrong.

I don't think there's anything very original in what I have written above. Take a bit of Clower and Yeager and stir.

Keynesians take disequilibrium seriously, and monetarists take money seriously. This post belongs in the somewhat narrow intersection of those two sets.

Perhaps what's wrong with macroeconomics is that fundamental issues like Say's Law need to be debated in blogs. Does it get debated anywhere else nowadays?