My Quantity Theory of ... WHATEVAH!!!
Well I was gonna respond to KNZN over in his comments, but then I thought, hey! Why write stuff up on someone else's blog when I can post it here and feel like I've done my blogging duty for the next few days or so.
So KNZN says he doesn't understand the Quantity Theory of Money because he doesn't understand Velocity of money. Fair enough, it's one of those things that seem deceptively simple yet actually pack in a lot of intellectual powder. I'm not so sure I fully understand it either.
So first, let's get nicky picky.
MV=PY
where
M = Money
V = Velocity of Money
P = Price Level
Y = Real GDP
is the Equation of Exchange. It is an identity which says that the number of transactions in an economy equals... wait for it, wait for it,... the number of transactions in an economy! (Measured as Real GDP).
Strictly speaking the Quantity Theory, as distinct from Equation of Exchange, is a behavioral proposition - that if M changes than the adjustment to keep the identity an identity, at least in the long run, takes place through P. This is QT as laid out by Hume, Copernicus (English language copies of this are actually pretty rare), and, if you read him charitably, Suetonious, the Roman dude (the story about Julius Ceasar looting Egypt and increasing the money stock back in the City of the Seven Hills).
But what is Velocity? To quote KNZN:
Velocity is typically defined as something like “the number of times an average dollar changes hands.”
and in simplest terms:
If all our money took the form of dollar bills, and if we placed a check mark on a dollar bill every time we used it to purchase value added, then the velocity of money would be the number of check marks added to the average dollar bill in a year.
But he says
We don’t have just one denomination of currency; we have many. Most of our transactions are electronic, anyhow. And we don’t put check marks on the currency when we use it.
I say: Yeah but that's not what matters.
If there's only one 100$ bill in the economy which changes hands 3 times per year, and 20 1$ bills, each of which changes hands 20 times per year then:
Money supply is 320$
Nominal GDP = PY = # of transactions = 320$
Which leaves V. Welp, if the 100$ turns over 3 times and each of the 100 1$ bills turns over 20 times than Velocity will be (20+100*3)/101=320/101.
And so we get MV=PY.
Does that seem sketchy to you? Well, it should because the relationship MV=PY is an identity, a tautology. That is we pretty much defined Velocity as that which is needed to make it add up. Let's leave that alone for a moment - the point is that different denominations are not the problem here. Neither is the ability to "check" each dollar bill as it circulates around (and even there, they do have serial numbers). Just like we don't observe every value-added transaction but still manage to come up with a number called GDP, or Y, we don't need to check every bill everytime it changes hands. There are indirect methods of calculating V, the most obvious one being of just calculating PY/M once we can get our definitions of P, M and Y straight (i.e. we don't NEED to calculate V independently). And that's where the trouble comes in.
P is basically the GDP deflator, the ratio of nominal to real GDP. But in calculating "real" GDP we got to settle on a method of choosing a base year, or a weight to average (chain) years and so on. This is gonna throw things of kilter somewhat.
Y (once you have P) is probably the easiest, in some sense, to calculate, by either adding up total expenditure, total incomes, or total value added. If these methods give similar numbers we're getting close. Still it's not a perfect measure.
And so we get to M, which is where all the action and trouble is. The problem with the Equation of Exchange as an identity is not the definition of V (and the fact that we have bills of different denominations does not matter - at least not any more than that we have different goods aggregated to something called Y and their prices to something called P) but the definition of M.
Since the Equation is a statement about transactions, pretty much anything used to make an exchange can be considered money. If I take off my right shoe and somehow manage to convince the bartender at my local pub to trade me a pint of ale for it, then both the shoe and the ale can be considered "money" - they are being used as a Medium of Exchange which is one of the functions of money. However, we generally also want Money to be a numeraire, a Unit of Account, so that we are able to express all the different transactions that take place in a complex economy in common units (dollars, ducats, wampums, cigarettes, big stones up on the hill over yonder). Shoes and ales are not in the same units. Dollars are. So we need to define more precisely by what we mean by Money. This is where all the different definitions of Money come in: M1 as currency plus checkable deposits, M2 as M1 plus savings and time deposits, ... M10 which I believe includes famous works of art, up to M-whatevah! at which point ales and right foot shoes are included. (Note that liquidity - the ease of carrying out transactions - declines with the index. Actually, it's the marginal liquidity which declines, to be precise).
But, as the appropriately named MVPY comments Each M (M1, M2..) have different velocities. This of course comes from dividing PY by a different definition of money and calculating out the needed V. Then you can take logs, differentiate with respect to time and see how this V changes over time. A good portion of why Milton Friedman insisted on M2 as the proper definition of money (If I recall my readings from some years ago correctly) is that that was the M2 which had the smallest changes in V, which fit nicely into the monetarist framework (basically, if V changes a lot the Fed should target interest rates, if not, it should target the money supply, which is what Friedman advocated).
But all the above is a discussion about The Equation of Exchange, NOT the Quantity Theory of Money. There's nothing in there about whether when M changes the variable that does the adjusting is P or M or even V. So even a simple identity can be a lot of trouble - once we start arguing about the Quantity Theory proper it will of course be even worse. I'm just glad I got Copernicus along with Friedman on my side in saying that inflation IS, for all practical purposes, always and everywhere a monetary phenomenon.
---
The second part of KNZN's post actually looks like an independent discovery of the Cambridge-k idea of money demand. You can take the equation of exchange and express it as:
M/P=Y/V
This is still an indentity, engineering. Now we add in the economics. We call M/P the "real balances" and let V=1/k(r) where k is an increasing function of the interest rate. Then we call Y*k(r)=L(Y,r) the liquidity function (or demand for liquidity). Hell, let's just say k=r. Then we get the Principles of Macro M/P=Y/r:
So
MV=PY is the Equation fo Exchange, an identity.
The Quantity Theory is the proposition that at least in the long run, it is the P which adjusts to changes in M.
The Cambridge theory of money demand is that Y/V=L(Y,r) is a money demand function, which in equilibrium, has to equal available real balances (money supply).
QT and CTofMd are based, and derived fropm the EoE, but are not synonymous with it nor are they necessary implications of it, nor are they the same thing.
Okay. Now we can argue about whether P adjusts or Y adjusts to changes in M, and what is the proper definition of money, and how exactly did those big-stones-up-on-the-hill-over-yonder function as money, and why is it that no economic model, neoclassical, heterodox, Austrian, whatever, has a decent theory of money
(obvious problem with the Clower Cash-in-Advance constraint way of introducing money into GE is that it restricts the available number of transactions, hence it cannot be welfare improving. So why have money?. Obvious problem of the Sidrauski Money-in-the-utility-function path is that it's ad hoc and a "reduced form" way to go at best. Anyway, apparantly under fairly general conditions they're the same things and generate "right" behavior, so maybe they're in the "crazy assumption, but works well nonetheless" category, like perfect competition).
But I'll save those arguments for another time, because I have to go and I need to somehow get my right shoe back.
(which is also when I'll add more relevant links to this post)
So KNZN says he doesn't understand the Quantity Theory of Money because he doesn't understand Velocity of money. Fair enough, it's one of those things that seem deceptively simple yet actually pack in a lot of intellectual powder. I'm not so sure I fully understand it either.
So first, let's get nicky picky.
MV=PY
where
M = Money
V = Velocity of Money
P = Price Level
Y = Real GDP
is the Equation of Exchange. It is an identity which says that the number of transactions in an economy equals... wait for it, wait for it,... the number of transactions in an economy! (Measured as Real GDP).
Strictly speaking the Quantity Theory, as distinct from Equation of Exchange, is a behavioral proposition - that if M changes than the adjustment to keep the identity an identity, at least in the long run, takes place through P. This is QT as laid out by Hume, Copernicus (English language copies of this are actually pretty rare), and, if you read him charitably, Suetonious, the Roman dude (the story about Julius Ceasar looting Egypt and increasing the money stock back in the City of the Seven Hills).
But what is Velocity? To quote KNZN:
Velocity is typically defined as something like “the number of times an average dollar changes hands.”
and in simplest terms:
If all our money took the form of dollar bills, and if we placed a check mark on a dollar bill every time we used it to purchase value added, then the velocity of money would be the number of check marks added to the average dollar bill in a year.
But he says
We don’t have just one denomination of currency; we have many. Most of our transactions are electronic, anyhow. And we don’t put check marks on the currency when we use it.
I say: Yeah but that's not what matters.
If there's only one 100$ bill in the economy which changes hands 3 times per year, and 20 1$ bills, each of which changes hands 20 times per year then:
Money supply is 320$
Nominal GDP = PY = # of transactions = 320$
Which leaves V. Welp, if the 100$ turns over 3 times and each of the 100 1$ bills turns over 20 times than Velocity will be (20+100*3)/101=320/101.
And so we get MV=PY.
Does that seem sketchy to you? Well, it should because the relationship MV=PY is an identity, a tautology. That is we pretty much defined Velocity as that which is needed to make it add up. Let's leave that alone for a moment - the point is that different denominations are not the problem here. Neither is the ability to "check" each dollar bill as it circulates around (and even there, they do have serial numbers). Just like we don't observe every value-added transaction but still manage to come up with a number called GDP, or Y, we don't need to check every bill everytime it changes hands. There are indirect methods of calculating V, the most obvious one being of just calculating PY/M once we can get our definitions of P, M and Y straight (i.e. we don't NEED to calculate V independently). And that's where the trouble comes in.
P is basically the GDP deflator, the ratio of nominal to real GDP. But in calculating "real" GDP we got to settle on a method of choosing a base year, or a weight to average (chain) years and so on. This is gonna throw things of kilter somewhat.
Y (once you have P) is probably the easiest, in some sense, to calculate, by either adding up total expenditure, total incomes, or total value added. If these methods give similar numbers we're getting close. Still it's not a perfect measure.
And so we get to M, which is where all the action and trouble is. The problem with the Equation of Exchange as an identity is not the definition of V (and the fact that we have bills of different denominations does not matter - at least not any more than that we have different goods aggregated to something called Y and their prices to something called P) but the definition of M.
Since the Equation is a statement about transactions, pretty much anything used to make an exchange can be considered money. If I take off my right shoe and somehow manage to convince the bartender at my local pub to trade me a pint of ale for it, then both the shoe and the ale can be considered "money" - they are being used as a Medium of Exchange which is one of the functions of money. However, we generally also want Money to be a numeraire, a Unit of Account, so that we are able to express all the different transactions that take place in a complex economy in common units (dollars, ducats, wampums, cigarettes, big stones up on the hill over yonder). Shoes and ales are not in the same units. Dollars are. So we need to define more precisely by what we mean by Money. This is where all the different definitions of Money come in: M1 as currency plus checkable deposits, M2 as M1 plus savings and time deposits, ... M10 which I believe includes famous works of art, up to M-whatevah! at which point ales and right foot shoes are included. (Note that liquidity - the ease of carrying out transactions - declines with the index. Actually, it's the marginal liquidity which declines, to be precise).
But, as the appropriately named MVPY comments Each M (M1, M2..) have different velocities. This of course comes from dividing PY by a different definition of money and calculating out the needed V. Then you can take logs, differentiate with respect to time and see how this V changes over time. A good portion of why Milton Friedman insisted on M2 as the proper definition of money (If I recall my readings from some years ago correctly) is that that was the M2 which had the smallest changes in V, which fit nicely into the monetarist framework (basically, if V changes a lot the Fed should target interest rates, if not, it should target the money supply, which is what Friedman advocated).
But all the above is a discussion about The Equation of Exchange, NOT the Quantity Theory of Money. There's nothing in there about whether when M changes the variable that does the adjusting is P or M or even V. So even a simple identity can be a lot of trouble - once we start arguing about the Quantity Theory proper it will of course be even worse. I'm just glad I got Copernicus along with Friedman on my side in saying that inflation IS, for all practical purposes, always and everywhere a monetary phenomenon.
---
The second part of KNZN's post actually looks like an independent discovery of the Cambridge-k idea of money demand. You can take the equation of exchange and express it as:
M/P=Y/V
This is still an indentity, engineering. Now we add in the economics. We call M/P the "real balances" and let V=1/k(r) where k is an increasing function of the interest rate. Then we call Y*k(r)=L(Y,r) the liquidity function (or demand for liquidity). Hell, let's just say k=r. Then we get the Principles of Macro M/P=Y/r:
So
MV=PY is the Equation fo Exchange, an identity.
The Quantity Theory is the proposition that at least in the long run, it is the P which adjusts to changes in M.
The Cambridge theory of money demand is that Y/V=L(Y,r) is a money demand function, which in equilibrium, has to equal available real balances (money supply).
QT and CTofMd are based, and derived fropm the EoE, but are not synonymous with it nor are they necessary implications of it, nor are they the same thing.
Okay. Now we can argue about whether P adjusts or Y adjusts to changes in M, and what is the proper definition of money, and how exactly did those big-stones-up-on-the-hill-over-yonder function as money, and why is it that no economic model, neoclassical, heterodox, Austrian, whatever, has a decent theory of money
(obvious problem with the Clower Cash-in-Advance constraint way of introducing money into GE is that it restricts the available number of transactions, hence it cannot be welfare improving. So why have money?. Obvious problem of the Sidrauski Money-in-the-utility-function path is that it's ad hoc and a "reduced form" way to go at best. Anyway, apparantly under fairly general conditions they're the same things and generate "right" behavior, so maybe they're in the "crazy assumption, but works well nonetheless" category, like perfect competition).
But I'll save those arguments for another time, because I have to go and I need to somehow get my right shoe back.
(which is also when I'll add more relevant links to this post)
posted by YouNotSneaky! at 7:21 PM
18 Comments:
Huh... isn't Y = real GDP, so that PY = nominal GDP?
And then there's the wacky PT version where T = no. of transactions... And the PQ version. Oi!
What I always understood by "QToM" is long-run neutrality and short-run non-neutrality.
Re: decent theory of money...
We don't have an empirical hypothesis concerning monetary aggregates but as theory, people figures out money long ago: double coincidence of wants, store of value, etc.
Ooops, typo fixed.
The problem with all them theories is that there's always alternatives:
Money is not the only way to store value, and in many circumstances not even the best one. As far as double coincidence of wants, yeah, that's probably what it's about but I haven't seen a convincing model of it.
And anyway, double coincidence of wants is just a coordination problem. Another way to solve is to centralize trade (i.e. build a city), which is why it's not needed in Walrasian GE.
It sounds like you’re saying that the equation MV=PT is just a way of stating the definition of velocity. If that’s the case, then the question is, “Why should I care about velocity?” I have a pretty good intuitive sense of why I care about price and quantity indexes. Between Keynes and Hicks, I’ve got a pretty good explanation of why I should care about the quantity of money, leaving aside for the moment the question of what exactly it is. But why should I expect the velocity of anything to have any interesting properties? (It might happen to have an interesting property because of the structure of the money demand function, but then why bother with the concept of velocity? Why not just talk about money demand?)
You should care because it's all about which variable does the adjustin'. If M changes then it could be P which changes, or it could be that Y changes, or it could be something else that changes. Just because we call that something else "velocity" does not mean that you should stop carrying about it. This is the oldest controversy in macroeconomics.
It's like Walras' Law of Caring. If there's n relevant variables, and you care about (n-1) of them, then you implicitly care about the nth one as well.
I agree though that it makes more sense in the context of a money demand function --> V, whatever it is, determining the shape as well as other variables (like interest rates) that the function depends on.
What I’m saying is, if I didn’t have a theory of money demand, I would assume that V always does the adjusting, because it’s an arbitrary definition and doesn’t provide any theory linking M with P and Y. By analogy, I could take the variable R = “inches of rainfall” and define the “effectiveness of rainfall” E with the equation RE = PY. Without a theory linking R to PQ, E is a useless definition, and I would assume that when R changes, E automatically adjusts to keep PQ constant (unless some other exogenous thing happens at the same time). When (as in the case of money) I do have a theory, the equation is superfluous because I can discuss everything in terms of the actual theory. So why ever bother with the concept of velocity?
The question of what values can V take is very interesting.
I tend to think of V as determined by technological (e.g. introduction of credit card technology) and institutional factors (e.g. banking practices).
Kaldor took an extreme position regarding he ability of the banking sector to offset monetary policy by changes in V. I find that highly unlikely.
I don’t see any a priori reason to place any restrictions of the values V can take. The burden of proof is on you if you disagree with Kaldor. You can probably meet that burden with a theory of money demand, but then why bother with the V concept in the first place? Why not just say, “There’s M, and then there’s PQ, and we have a money demand theory that says what the relationship will be.”?
It’s not clear to me why technology is an issue (except that it affects money demand). Compare two economies that use purely paper money. In economy #1, each person holds 100 one-dollar bills; in economy #2, each person holds 10 ten-dollar bills, so M is the same. Let’s say Y is the same too, but in economy #2, the price level is 10 times higher. Everything in economy #1 costs $1, and everything in economy #2 costs $10. When you want to buy something, you hand a single bill to the vendor: same technology in both economies (unless you consider printing a zero to be a technological advance), but V is 10 times higher in economy #2. The only practical difference is that, in economy #2, your real money balances are smaller. But unless you have a theory of money demand, we don’t know why anyone should care about real money balances.
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The Quantity Theory of Money is obviously not for the faint hearted or those with non-mathematical brains - but impressive nevertheless!
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It is indeed an interesting theory and concept, thanks for the explanation, it is very good.
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