I used to think that the Equation of Exchange, MV=PY, was just a different way of writing the Cambridge Equation, M=kPY, with V=1/k. And of the two I preferred the M=kPY formulation, because it looked more like a money supply=money demand condition, and reminded us that V=1/k must be interpreted as desired velocity, if MV=PY was to be more than just an identity, and desired velocity could be a function of many things. Plus, MV=PY was just so peculiar a way to write a supply=demand equilibrium condition.
Now I've changed my mind. The reasons I now like MV=PY are the very same reasons I used to dislike MV=PY. Money is peculiar, so MV=PY needs to be peculiar as well. And actual velocity is just the average of individuals' desired velocities.
I can imagine writing an equilibrium condition for houses, or jewelry, as M=kPY. The (nominal) demand for houses or jewelry is proportionate to nominal income. And in equilibrium the demand for houses or jewelry must equal the supply, M, so that the stock of houses or jewelry is willingly held. But I cannot imagine writing the equilibrium condition for houses or jewelry as MV=PY. What has the velocity of circulation of houses, or jewelry, got to do with anything, except realtors' and jewelers' commissions?
Spending 3 years' income on a house and 2 months' income on a ring doesn't mean you should change houses every 3 years and rings every 2 months.
Money is peculiar, and it is only fitting that we write the equilibrium condition for money in a peculiar way, especially in a way that reminds us of money's peculiarity. Money is a medium of exchange. It circulates. It flows into and out of our hands every time we sell or buy something else. And we demand it precisely because it is a medium of exchange.
I can make my actual equal my desired velocity of circulation of money in a way I cannot always make my actual equal my desired stock of houses or jewelry.
Suppose my desired stock of houses or jewelry increases. I have an excess demand for houses, or jewelry. There is only one action I can take to satisfy my excess demand for houses or jewelry: go out and buy more houses or jewelry. Money is peculiarly different because, unlike houses and jewelry, money is flowing into and out of my pocket every month. Suppose my desired stock of money increases. I have an excess demand for money. There are two actions I could take to satisfy my excess demand for money: I could go out and buy more money (sell more of something else), or I could sell less money (buy less of something else).
The first way of getting more houses, jewelry, or money, might fail. It can only succeed if I can find a willing seller of houses, jewelry, or money. (A willing seller of money means a willing buyer of whatever it is I want to sell to get more money).
The second way of getting more money (by buying less of something else) is peculiar to money. And it can't fail. I don't need to find a willing trading partner to buy less of the things I normally buy. I just stop buying them. Or, at least, it can't directly fail. It can't fail for me as an individual; but it can fail indirectly if everyone tries to do the same thing.
Suppose everyone has an excess demand for houses or jewelry. Everyone tries to buy more jewelry or houses, but fails because they can't all find willing sellers. One of two things then happens: either the price of houses or jewelry rises enough to eliminate the excess demand; or else they can't satisfy their excess demands, and give up.
Suppose everyone has an excess demand for money. Remember the two ways of getting more money. If everyone tries the first way, and tries to buy more money (sell more other things), they will fail directly, because they won't find willing sellers of money (buyers of other things). One of two things happens: either the price of money rises enough to eliminate the excess demand; or else they can't satisfy their excess demand this way. But if they fail to satisfy their excess demand for money this first way, they won't just give up, they will try something else instead. They will naturally try the second way of getting more money: sell less money (buy less other things).
The second way of getting more money cannot fail at the individual level. Nobody can stop you buying less of other things. You don't need to find a willing partner. It will fail at the aggregate level, because one person's reduced outflow of money is another person's reduced inflow of money. But it won't fail at the individual level, so individuals will not give up trying to satisfy their excess demand for money. They will keep on trying, and the reduced inflow of money only makes them try harder to reduce the outflow of money. Until eventually, income falls so much that they are satisfied with their existing stocks of money.
Another way of saying the same thing, much more simply, is that each individual can choose whatever velocity of circulation he wants, regardless of what others want. And aggregate actual velocity is just the weighted sum of those individual choices. The same is not true for individuals' demands for houses and jewelry.
What I have just described is a monetarist version of the Keynesian paradox of thrift, and the monetarist version is much more accurate.
The paradox of thrift is a misnomer. "Thrift" means saving, and "saving" in Keynesian national income accounting means anything you do with your income from newly-produced goods except spend it on newly-produced consumption goods. It is only savings in the form of the medium of exchange that can lead to the so-called paradox of thrift.
A shift of household demand away from newly-produced consumption towards newly-produced investment goods is an example of increased desired savings, but it leads to no paradox of thrift.
A shift of household demand away from newly-produced consumption towards antique furniture is an example of increased desired savings, but it is unlikely to lead to a paradox of thrift. One of two things will happen: either the price of antique furniture rises enough to eliminate the excess desired savings; or it doesn't, and there is an excess demand for antique furniture that cannot be satisfied. Unable to buy more antique furniture, people will give up, and try to buy something else instead. Probably newly-produced furniture.
A shift of household demand away from newly-produced consumption towards holding more of the medium of exchange is an example of increased desired savings that will lead to a multiplier decline in output (unless of course the price of money rises or the supply of money increases).
And a shift of household demand away from newly-produced consumption towards holding more bonds will only cause a paradox of thrift if the price of bonds rises so the rate of interest falls and leads to a decline in desired velocity. If desired velocity stayed the same, the extra supply of loans would find willing borrowers who would spend the flow of money that households don't want to spend. If the increased demand for antique furniture lead indirectly to an increased demand for money ("we can't find any antiques to buy, or they are so expensive, so let's just hold cash while we're waiting") then it would also cause the paradox of thrift.
The Keynesian multiplier, IS curve, and involuntary unemployment only make sense in a monetary exchange economy, where money is the medium of exchange, and the unemployed cannot easily barter their services with each other. MV=PY is the way monetarists remind themselves (or should remind themselves) that we live in a monetary exchange economy. Money deserves a peculiar formula because the medium of exchange is peculiar. In a monetary exchange economy with n goods plus the medium of exchange, there are n markets. Money appears in each of those n markets; each other good only appears once, in its own market. There are n excess demands (or supplies) for goods, one each, and n excess demands (or supplies) for money. (And the only valid formulation of Walras' Law is one that recognises all n excess demands for money.) If you want more houses, or jewelry, there is only one market where you can try to buy more, if you can find a willing seller. If you want more money there are n markets in which you can either try to buy more, or always succeed in selling less.