What we are talking about when we talk about velocity

Economists of my generation recall a long boring debate over monetary policy, which frequently involved the term ‘velocity.’  Monetarists claimed the Fed should stop trying to fine tune the economy, and instead keep the money supply growing at a steady rate of about 4%.  Keynesians countered that velocity was unstable, so the policy would not work.  Monetarists countered that much of the instability of velocity was due to instability of money, and that if money growth was more stable, velocity would also be more stable.

And so the debate raged on and on.  So here’s my question; why the heck does it matter what happens to velocity?  Why should we even care?  Why did the Keynesians work so hard to show velocity would be unstable, and why did the monetarists fight so hard to show it would be stable if only we would target M?  Why?

You might think the reason is obvious—just look at the famous equation of exchange:

M*V=P*Y

If velocity is stable, then a steady growth of M would yield a steady growth of P*Y.  Yes, I get that.  When we talk about the equation of exchange we are just talking about definitions.

Obviously in the long fruitless debate over velocity during the 1950s, 1960s, and 1970s, the implicit benchmark for a successful monetary policy was stable growth in P*Y.  That’s obvious.

But here’s the mystery, almost no one in the 1950s, 1960s, or 1970s was actually advocating NGDP targeting!  People have dug up quotations that seemed to show Milton Friedman opposed NGDP targeting.  I don’t believe them.  Friedman supported NGDP targeting, whether he knew it or not.  Tell me which of the following imaginary scenarios would have been more likely:

Scenario A:  Research using a sophisticated MIT supercomputer shows the Keynesians were wrong and Milton Friedman was right all along.  Fluctuations in M2 velocity were 100% caused by fluctuations in M2 itself.  That’s why the collapse of M2 in the early 1930s was accompanied by sharply falling velocity.  In addition, the study showed that M2 velocity would be virtually constant if M2 were to grow at a steady rate of 4%/year.  Friedman responds with a note of exultation.  “Finally I’ve been proved right, and by an MIT study no less.  There is no longer any reason to refrain from putting the Fed on automatic pilot, replacing Fed officials with a computer.  No more discretion.”

Scenario B:  Research using a sophisticated MIT supercomputer shows the Keynesians were wrong and Milton Friedman was right all along.  Fluctuations in M2 velocity were 100% caused by fluctuations in M2 itself.  That’s why the collapse of M2 in the early 1930s was accompanied by sharply falling velocity.  In addition, the study showed that M2 velocity would be virtually constant if M2 were to grow at a steady rate of 4%/year.  Friedman is dismayed by the findings.  “Oh my God, that this means that if the Fed targeted the money supply growth rate at 4% a year, we’d have steady 4% growth in NGDP.  That would be horrible, as it means that if RGDP fell by 1% then we’d have 5% inflation.  I now see the folly of my ways; the Fed must not increase the money supply at a constant rate, but rather must use discretion to prevent the evils that would flow from steady NGDP growth .”

Believe it or not, I’ve had commenters who implicitly claim scenario B is more plausible.

For three decades economists debated monetary policy using the implicit assumption that stable NGDP growth is the benchmark for success, that NGDP targeting is optimal.  And yet for some reason they never figured that out.  Almost no one was advocating NGDP targeting from 1950-1980.


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71 Responses to “What we are talking about when we talk about velocity”

  1. Gravatar of John John
    6. July 2012 at 06:34

    Velocity= distance/time. Leave the physics to physicists.

  2. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 06:52

    The theory that a constant growth in P*Y will occur if growth in M is constant, with the assumption that V would be stable, has been empirically falsified in the US, and in Australia.

    In the US, MZM velocity has been decreasing since the early 1980s despite there being no secular decline in MZM growth, and M2 velocity has been decreasing since the mid 1990s despite there being no secular decline in M2 growth.

    In Australia, M2 growth accelerated, and yet NGDP remained stable, which implies a falling V as well.

    Notice especially the period starting mid to late 1990s, for both charts. I’ll leave it to the reader to figure out what happened…

  3. Gravatar of 123 123
    6. July 2012 at 07:03

    Scott,
    Here is one treatment of velocity:
    http://www.ecb.int/press/pr/date/1998/html/pr981201_3.en.html

  4. Gravatar of Jonathan M.F. Catalán Jonathan M.F. Catalán
    6. July 2012 at 07:09

    The fact that major fluctuations in the demand for money (what we’re really talking about when we mention ‘V’) tend to correlate with fluctuations in the supply of money doesn’t imply any direct causation. Rather, what if both the fluctuation in supply and the fluctuation in demand are caused by something else, like real factors? (A rise in unproductive debt?) This would also explain uneven correlations between changes in the supply of money and changes in V (and, thus, would explain imperfect correlation between changes in expectations and changes in the supply of money).

  5. Gravatar of RPLong RPLong
    6. July 2012 at 07:13

    I find Hazlitt’s treatment very clarifying:
    http://mises.org/daily/2916

  6. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:17

    John:

    Velocity= distance/time. Leave the physics to physicists.

    Years ago one of my econometrics profs mentioned an economics methodology called “particle filtering.”

    Most economists have physics envy, John. They’ve had it since the 1950s, when Friedman played an important role in convincing the majority of economists that they can do economics by mimicking the physicists.

    It’s why the majority of economists are empiricist-positivists (what they preach of course; not necessarily what they practise).

    Too bad that the presupposition of empiricist-positivism, of time invariance in the operation of causes as such, does not and cannot apply to knowledge and action.

    ——–

    The belief that stable growth in M leads to stable growth in P*Y, is a belief that is derived from assuming time invariance in the operation of causes with respect to knowledge and actions, hence its fallaciousness.

  7. Gravatar of Mark A. Sadowski Mark A. Sadowski
    6. July 2012 at 07:20

    Scott,
    Noah’s article criticizing NGDP futures became one of the links at Economist’s View today:

    http://economistsview.typepad.com/economistsview/2012/07/links-for-07-06-2012.html

    Since he admits that he didn’t do his homework shouldn’t he include an update?

  8. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:23

    RPLong:

    I find Hazlitt’s treatment very clarifying:

    FTA:

    “It [MV=PT] seems to have had a wonderful longevity as a result of its apparent simplicity, its apparent mathematical precisions — and because it is easier to teach than other theories.”

    That pretty much sums it up. Time invariance in the operation of causes with respect to the money supply is just…easier to think about, hence its popularity.

  9. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:29

    “To begin with, the “velocity of circulation” of money is a misnomer. It is simply a figure of speech, a metaphor — and a misleading one. Strictly speaking, money does not “circulate”; it is exchanged against goods. Money is said to “circulate” because it changes hands, or, more precisely, changes ownership. But when a house, say, frequently changes ownership we do not say that it “circulates.” If we are to apply the metaphor of circulation to money, then we should also logically apply it to goods. For money (except in borrowing or in paying off debts) is always exchanged against goods (or services). Therefore the “velocity of circulation” of money can never be any greater than the “velocity of circulation” of goods.”

  10. Gravatar of Jeremy Goodridge Jeremy Goodridge
    6. July 2012 at 07:33

    It’s even worse than what you say. Friedman later acknowledged that we should move money to ACCOMMODATE changes in velocity. So, that seals the deal — he was interested in stabilizing NGDP.

    Here’s an interview with Bob Kuttner in 2005 in which he talks about velocity :

    http://prospect.org/article/interview-milton-friedman

    And here’s a relevant quote::

    RK: But in the context of a stock market collapse, doesn’t that take second place to the objective of keeping adequate liquidity in the system.

    MF: No, In order to keep the general price level stable, you need to keep adequate liquidity in the system..

    RK: Yes

    MF: You have to offset velocity—the stock market will affect velocity. And that is to say, when a stock market crashes, there is associated a demand for real balances, and that is equivalent to a decline in velocity. So you have to increase the money supply to offset that decline in velocity. If all you want is stability of the general price level, apparently it’s an easy job to do. I really must say I am surprised at how well all of these different countries have been able to keep to their inflation targets.

  11. Gravatar of Alex Godofsky Alex Godofsky
    6. July 2012 at 07:33

    It would actually be quite reasonable to examine and talk about the circulation of houses.

  12. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:39

    “A common classroom illustration runs something like this: A owes B a dollar, who owes C a dollar, who owes D a dollar, who owes E a dollar, who owes A a dollar. If they sit around a table, and each pays the dollar he owes to the other, then the dollar “circulates,” and one dollar “does the work of” five dollars. Two things may be noticed about this illustration. First, in the situation described, no actual dollar would have to change hands at all: debts could be cancelled out by mere bookkeeping transactions. This mutual cancellation of debts occurs in actual practice every day, and on a large scale, in bank clearing houses, or institutions that act as clearing houses. Secondly, the illustration, in fact, applies only to borrowing, or to paying off previously incurred debts.

    “But what we have to deal with, in the so-called circulation of money, is the exchange of money against goods. Therefore V and T cannot be separated. Insofar as there is a causal relation, it is the volume of trade which determines the velocity of circulation of money rather than the other way around.

    “What the mathematical quantity theorists seem to forget is that money is not exchanged against a vacuum, nor against other money (except in bank clearings and foreign exchange), but against goods. Hence the velocity of circulation of money is, so to speak, merely the velocity of circulation of goods and services looked at from the other side. If the volume of trade increases, the velocity of circulation of money, other things being equal, must increase, and vice versa.”

    —–

    Oh how often do economists divorce economic concepts from their praxeological foundation and manipulate and rearrange them in a purely idealistic world of free thought.

    It makes it almost a guarantee that person who forms new conceptualizations in this way will fall into the trap of hypostatization (i.e. to ascribe substance or real existence to mental constructs or concepts).

  13. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:42

    Alex Godofsky:

    It would actually be quite reasonable to examine and talk about the circulation of houses.

    Really? Circulation implies a movement, a dynamic phenomena. Houses don’t move. People move in and out of houses. It’s why we say “Mr/Mrs Smith moved out”, rather than “The house moved away from Mr/Mrs Smith.”

    How many economists actually talk about the circulation of houses? I’ve never heard it before. Have you?

  14. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:55

    “Cash holdings are sometimes greater and sometimes smaller with the same individual. But nobody ever has cash holdings greater than he wants to have. If he thinks that his cash holding are excessive, he invests the surplus either by buying (producers’ or consumers’ goods) or by lending it. (Time deposits are one method of lending money.) It is a judgment of value to call somebody’s cash holding “hoarding.” The individual concerned believes that under the given state of affairs, the best policy (let us say: the minor evil) is to increase his cash holdings. It does not matter whether I approve of his behavior or not. His behavior — not my subjective opinion about its expediency — is a factor influencing the formation of market prices.”

    “It is futile to distinguish between “circulating” money and “idle” money. Money changes hands without being ownerless for any fraction of time. Money may be in the process of transportation, traveling in railroad cars or in other means of transportation. But it is, even from the legal point of view, always in somebody’s possession.” – Mises in a letter to Hazlitt.

    How many times are our progressive and monetarist friends conflating their value judgments for objective economic principles binding on the rest of us? As if their value judgments concerning other people’s waiting times for holding money, are so enlightened that they are to be imposed on everyone!

    This is what they think: “Holding money is fine, but WE PROGRESSIVES reserve the right to decide when your holding of money turns into the evil “hoarding” of money, and that subjective value judgment from US is to annihilate your subjective value judgment concerning your own money. When OUR subjective valuation decides that your subjective judgment concerning your own waiting time for your own money is not to OUR liking, that is justification for the central bank print money for its friends, to devalue your cash balance, so that “somebody” will do their social bound duty of “spending”, so that there will be enough “spending” in “society” according to OUR wishes. The individual’s voluntary choice is to be annihilated by the will of “society” (i.e. the state).”

  15. Gravatar of ssumner ssumner
    6. July 2012 at 07:55

    123 and RPLong, Thanks for the links.

    Jonathan, Sure, there are lots of possibilities with causation. The question is what does it all mean?

    Jeremy, Yes, but he really should have been talking about real money demand in that quote, not velocity.

  16. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 07:57

    “In a changing world everybody is under the necessity of keeping an amount of ready cash on hand. This desire creates the demand for money and makes people willing to sell goods and services in exchange for money. A realistic theory of the value and the purchasing power of money must therefore start from a recognition of these desires. The changes in the purchasing power of the monetary unit are brought about by changes arising in the relation between the demand for money, i.e., the demand for money for cash holding, and the supply of money.”

    The main deficiency of the velocity of circulation concept is that it does not start from the actions of individuals but looks at the problem from the angle of the whole economic system. This concept in itself is a vicious mode of approaching the problem of prices and purchasing power. It is assumed that, other things being equal, prices must change in proportion to the changes occurring in the total supply of money available. This is not true.”

    Approaching economic problems from the angle of the whole economic system, instead of the actions of individuals. Hmmmm….I wonder where we have seen that before?

  17. Gravatar of nanute nanute
    6. July 2012 at 08:07

    Scott,
    Is there a nexus between velocity and the M1 multiplier? Haven’t looked lately, but last time I did the multiplier was “negative.”

  18. Gravatar of Alex Godofsky Alex Godofsky
    6. July 2012 at 08:13

    MF:

    Really? Circulation implies a movement, a dynamic phenomena. Houses don’t move. People move in and out of houses. It’s why we say “Mr/Mrs Smith moved out”, rather than “The house moved away from Mr/Mrs Smith.”

    This is inane. Circulation of the ownership of houses among people. It’s an analogy to the physical circulation of a fluid, not a claim that houses are physically moving around.

  19. Gravatar of Jonathan M.F. Catalán Jonathan M.F. Catalán
    6. July 2012 at 08:28

    That excerpt from a letter from Mises to Hazlitt is strange, given that what Mises thinks (in Human Action0 matters in knowing whether an economy is going through a malinvestment boom is the volume of circulating credit.

  20. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 09:16

    Alex Godofsky:

    Circulation of the ownership of houses among people. It’s an analogy to the physical circulation of a fluid, not a claim that houses are physically moving around.

    Well, you said it was reasonable to say the circulation of houses. Now you’re saying circulation of ownership claims to houses. Those are I think you will agree two different things.

    OK, if you want to talk about circulation of ownership claims to goods, then that runs into the problem of whether the underlying goods are even constant or durable enough to warrant as a standard to give meaning to ownership claims circulating for “that good there.” It might work for durable goods like houses, but for prepared foods, say, it won’t make much sense.

  21. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 09:23

    Jonathan M.F. Catalan:

    That excerpt from a letter from Mises to Hazlitt is strange, given that what Mises thinks (in Human Action0 matters in knowing whether an economy is going through a malinvestment boom is the volume of circulating credit.

    Yes but Mises did not say that a given quantity of circulating credit has constant effects. There is no mechanical one to one relationship between volume of circulating credit and extent of malinvestment boom (however much one can even measure extent of a malinvestment boom).

    There is, I think, a difference between saying circulation credit brings about a boom, and saying 10% credit expansion relative to GDP generates an malinvestment boom the size of “X”.

    More importantly however, Mises held that the length and extent of the boom depends on the subjective valuations of entrepreneurs. An individual entrepreneur can gain from it as long as he gets out in time before others.

  22. Gravatar of Mark A. Sadowski Mark A. Sadowski
    6. July 2012 at 09:53

    Noah Smith says:
    “Well, Mark, read the excerpted quotes above. Many times, Sumner has explicitly referred to targeting “NGDP futures prices”. Therefore my critique applies exactly as written.

    It also raises the question of how Scott’s claim that “the NGDP futures price never fluctuates at all in my proposal” squares with the idea of targeting a futures price. I’m just waiting for Scott to respond and clarify.”

    http://noahpinionblog.blogspot.com/2012/07/excess-volatility-and-ngdp-futures.html?showComment=1341589104229#c5564424623605757832

  23. Gravatar of Alex Godofsky Alex Godofsky
    6. July 2012 at 10:15

    MF:

    Well, you said it was reasonable to say the circulation of houses. Now you’re saying circulation of ownership claims to houses. Those are I think you will agree two different things.

    No, I won’t, because only an idiot would read “circulation of houses” and think it referred to a giant flowing mass of houses tumbling around like balls in a playpen.

  24. Gravatar of Full Employment Hawk Full Employment Hawk
    6. July 2012 at 10:37

    Friedman, throughout most of his career, believed that volocity would be stable if the growth of M2 was stable, so that M x V would be stable. However some fluctuations in velocity would occur, but they would be minor, so that their effect on nominal GDP would be minor. Attempts to offset the minor fluctuations in velocity by changing the rate of growth of M would cause monetary authorities to make mistakes and would therefore result in causing greater fluctuations in M x V than a steady growth in M. Therefore attempts by policymakers to offset the fluctuations in V would make things worse and not better.

    The overwhelming proof that this stability in V does not exist has undermined traditional monetarism. The crucial way market monetarism differs from traditional monetarism is in asserting that the rate of growth of M x V needs to be held constant at some specific rate. Therefore “variable velocity monetarism” would be a better name for this approach to monetarism than “market monetarism.”

  25. Gravatar of Jonathan M.F. Catalán Jonathan M.F. Catalán
    6. July 2012 at 11:16

    Major Freedom,

    Okay, I wasn’t trying to suggest some type of relationship between the quantity of money in circulation and the extent of malinvestment. Rather, I’m bringing attention to the fact that at one point Mises did believe it relevant to look at “circulating” and “idle” credit, since the latter doesn’t play a role in the business cycle (until it begins to circulate).

  26. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 11:44

    Alex Godofsky:

    Well, you said it was reasonable to say the circulation of houses. Now you’re saying circulation of ownership claims to houses. Those are I think you will agree two different things.

    No, I won’t, because only an idiot would read “circulation of houses” and think it referred to a giant flowing mass of houses tumbling around like balls in a playpen.

    I think only an idiot would say houses, and ownership claims to houses, are interchangeable such that saying circulation of houses is the same thing as saying circulation of ownership claims to houses.

    I thought it was idiotic to say “circulation of houses” precisely because it conveys such an image of houses tumbling around.

    If I said circulation of air, what is moving? If I said circulation of cars, what is moving? If I said circulation of electric current, what is moving? That which follows “of” is what conveys the thing that is moving.

    ——

    JMFC:

    I’m bringing attention to the fact that at one point Mises did believe it relevant to look at “circulating” and “idle” credit, since the latter doesn’t play a role in the business cycle (until it begins to circulate).

    I think in the letter Mises was talking about “circulating” versus “idle” money balances, and how it is futile to distinguish between the two. Circulating credit I think should be distinguished from savings backed credit. I can’t even picture what “idle credit” would look like apart from it being a money balance.

  27. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 11:57

    Full Unemployment Hawk:

    The overwhelming proof that this stability in V does not exist has undermined traditional monetarism. The crucial way market monetarism differs from traditional monetarism is in asserting that the rate of growth of M x V needs to be held constant at some specific rate.

    V is in part determined by how many goods and services are exchanged against money.

    The number of goods and services exchanged is not determined by V, as if V is some exogenous driver of its own. V is in fact a fudge factor that is defined as the ratio of the other variables, namely V = P*Y/M. It has no meaning outside of this.

    If the Fed targets M*V, then they are of course targeting P*Y. So if production decreases, the Fed brings about higher price inflation, and if production increases, the Fed brings about lower price inflation.

    Therefore “variable velocity monetarism” would be a better name for this approach to monetarism than “market monetarism.”

    Why not variable M monetarism, since M can fluctuate up or down on the basis of M deflation? M is not just base money. M is aggregate money, which includes money created via credit expansion. Credit can deflate. If M*V fluctuates due to M fluctuating, then the Fed will have to change M until they hit the M*V target.

  28. Gravatar of John John
    6. July 2012 at 12:09

    M= Money
    V= velocity of money spent on potatoes
    P= price level of potatoes
    Y= total volume of potatoes purchased

    If I have $200 and I spend 1/20th of it on potatoes, then my spending on potatoes equals the price level of potatoes times the number of potatoes I purchase. This is really deep stuff.

    Here is the definition of velocity in this transaction. Velocity of money spent on potatoes equals amount of money spent on potatoes divided by the amount of money (V= P*Y/M).

    V= P*Y/M

    So therefore we can define velocity by averaging all the prices in the economy. To do this we can add up the price of an iPhone and a pencil then divide by an iPhone and a pencil. Then we find the total output by adding up all the stuff in the economy; if this seems problematic just add up the pounds of iPhones produced to the pounds of legal services produced. Then divide by the average total stock of money; something that every economist can agree on.

    I don’t see why velocity is such a debate.

  29. Gravatar of Major_Freedom Major_Freedom
    6. July 2012 at 12:34

    If I have $200 and I spend 1/20th of it on potatoes, then my spending on potatoes equals the price level of potatoes times the number of potatoes I purchase.

    True, but it’s better to say the price level of potatoes equals total spending on potatoes divided by the total supply of potatoes.

    I don’t see why velocity is such a debate.

    It’s a debate because it’s a fudge factor.

    M*V=P*Y is just saying money spent equals money received, where if aggregate demand changes apart from changes in the money supply, then we are to conclude that “velocity” changed. Velocity has no independent meaning apart from being defined as a relation between the other variables: P*Y/M. Price has a meaning independent from Y and M. Output has a meaning apart from P and M. Money supply has a meaning apart from P and Y. But velocity has no meaning apart from P, Y and M, hence the debate over V.

  30. Gravatar of Nick Rowe Nick Rowe
    6. July 2012 at 13:04

    John: “Velocity= distance/time. Leave the physics to physicists.”

    Yep. What economists call “Velocity” is really what is measured by a car’s tachometer, rather than its speedometer. It is revolutions per minute (or per year is a more useful unit, since we measure GDP in output per year). What is the correct word for what a tachometer measures? V has the units 1/time.

    Scott: “Obviously in the long fruitless debate over velocity during the 1950s, 1960s, and 1970s, the implicit benchmark for a successful monetary policy was stable growth in P*Y. That’s obvious.”

    Hmm. Yes. It is obvious, now you have said it. It wasn’t obvious (to me) before.

  31. Gravatar of RebelEconomist RebelEconomist
    6. July 2012 at 13:40

    I agree with John. When I studied the mathematics of physics, I was taught that velocity comprised both magnitude and direction. The idea of direction certainly does not apply to money velocity, which might be better described as a turnover rate. For money, it would be more appropriate to use the term “speed” rather than velocity, since speed is the rate of movement without regard to direction, but “speed” does not sound as scientific!

  32. Gravatar of Full Employment Hawk Full Employment Hawk
    6. July 2012 at 15:08

    “Velocity has no independent meaning apart from being defined as a relation between the other variables: P*Y/M.”

    That is true for realized velocity. But monetarism’s concept of velocity is desired velocity, how often people want to spend their money holdings during some period of tims, which is the reciprocal of the demand for money for a given level of P and Y (if we are talking about income velocity, rather than transactions velocity). In equilibrium actual and desired velocity are equal, out of equilibrium they are not. While only actual velocity can be readily measured, the theoretical work on velocity is with respect to desired velocity, which Friedman, during most of his career argued was a stable function of a few variables.

    For any given M, a change in desired velocity will cause people to spend their money more frequently, which will cause a change in P or Y or both. Friedman during most of his career argued that such changes in desired velocity are not a major factor in affecting P and Y.

    Your assertion above shows that you really do not have a good understanding of monetarism.

  33. Gravatar of Jeff Jeff
    6. July 2012 at 16:39

    The lack of timely, accurate and frequent data may be why no one has targets NGDP. The official GDP numbers are quarterly numbers that do not become available until long after the quarter is over, and even so they are subject to substantial revisions for some time afterward. Not only that, for some kinds of production what’s actually measured is real output, and then an appropriate price index has to be measured and applied to come up with nominal output for that sector.

    On the other hand, we have very good weekly data for money supply numbers available within a week or two after a week is over, and the revisions are much smaller than the GDP revisions are.

    If you want to target a nominal variable, you should give some consideration to these issues. Using futures prices to stand in for the unavailable data is a good idea, but then you’re going to have convince people that there’s no way for a big financial player to profitably rig the game.

  34. Gravatar of flow5 flow5
    6. July 2012 at 16:59

    Milton Friedman never knew what velocity was, nor how to measure it:

    The transactions concept of money velocity (Vt) has its roots in Irving Fisher’s equation of exchange (PT = MVt), where (1) M equals the volume of means-of-payment money; (2) Vt, the transactions rate of turnover of this money; (3) T, the volume of transactions units; and (4) P, the average price of all transactions units. The “econometric” people don’t like the equation because it is impossible to calculate P and T. Presumably therefore the equation lacks validity. Actually the equation is a truism – to sell 100 bushels of wheat (T) at $4 a bushel (P) requires the exchange of $400 (M) once, or $200 (Vt) twice, etc.

    The real impact of monetary demand on the prices of goods and serves requires the analysis of “monetary flows”, and the only valid velocity figure in calculating monetary flows is Vt. Milton Friedman’s Income velocity (Vi) is a contrived figure (Vi = Nominal GDP/M). The product of MVI is obviously nominal GDP. So where does that leave us? In an economic sea without a rudder or an anchor. A rise in nominal GDP can be the result of (1) an increased rate of monetary flows (MVt) (which by definition the Keynesians have excluded from their analysis), (2) an increase in real GDP, (3) an increasing number of housewives selling their labor in the marketplace, etc. The income velocity approach obviously provides no tool by which we can dissect and explain the inflation process.

    To the Keynesians, aggregate monetary demand is nominal-GDP, the demand for services (human) and final goods. This concept excludes the common sense conclusion that the inflation process begins at the beginning (with raw material prices and processing costs at all stages of production) and continues through to the end.

    We know that to ignore the aggregate effect of money flows on prices is to ignore the inflation process. And to dismiss the concept of Vt by saying it is meaningless (that people can only spend their income once) is to ignore the fact that Vt is a function of three factors: (1) the number of transactions; (2) the prices of goods and services; (3) the volume of M.

    Inflation analysis cannot be limited to the volume of wages and salaries spent. To do so is to overlook the principal “engine” of inflation – which is of course, the volume of credit (new money) created by the Reserve and the commercial banks, plus the expenditure rate (velocity) of these funds. Also overlooked is the effect of the expenditure of the savings of the non-bank public on prices. The (MVt) figure encompasses the total effect of all these monetary flows (MVt).

  35. Gravatar of flow5 flow5
    6. July 2012 at 17:14

    http://fraser.stlouisfed.org/docs/publications/bms/1914-1941/section5.pdf

    To those who may question the valildity of using transaction data with GDP data, there is evidence to prove that rates-of-change in nominal gDp can serve as a proxy figure for rates-of-change in all transactons. Rates-of-change in real gDp have to be used, of course, as a policy standard.

  36. Gravatar of flow5 flow5
    6. July 2012 at 17:17

    http://fraser.stlouisfed.org/docs/publications/bms/1914-1941/section5.pdf

    To those who may question the valildity of using transaction data with GDP data, there is evidence to prove that rates-of-change in nominal gDp can serve as a proxy figure for rates-of-change in all transactons. Rates-of-change in real gDp have to be used, of course, as a policy standard.
    Vt:
    1919 ,,,,,,, 24.5
    1920 ,,,,,,, 23.8
    1921 ,,,,,,, 20.8
    1922 ,,,,,,, 21.6
    1923 ,,,,,,, 20.8
    1924 ,,,,,,, 20.7
    1925 ,,,,,,, 21.7
    1926 ,,,,,,, 22.2
    1927 ,,,,,,, 23.4
    1928 ,,,,,,, 26.2
    1929 ,,,,,,, 29.9
    1930 ,,,,,,, 22.4
    1931 ,,,,,,, 18.1
    1932 ,,,,,,, 14.8
    1933 ,,,,,,, 15.3
    1934 ,,,,,,, 16
    1935 ,,,,,,, 15.8
    1936 ,,,,,,, 16.2
    1937 ,,,,,,, 16.1
    1938 ,,,,,,, 14
    1939 ,,,,,,, 13.6
    1940 ,,,,,,, 12.9
    1941 ,,,,,,, 14

  37. Gravatar of flow5 flow5
    6. July 2012 at 17:17

    http://fraser.stlouisfed.org/docs/publications/bms/1914-1941/section5.pdf

    To those who may question the valildity of using transaction data with GDP data, there is evidence to prove that rates-of-change in nominal gDp can serve as a proxy figure for rates-of-change in all transactons. Rates-of-change in real gDp have to be used, of course, as a policy standard.
    Vt:
    1919 ,,,,,,, 24.5
    1920 ,,,,,,, 23.8
    1921 ,,,,,,, 20.8
    1922 ,,,,,,, 21.6
    1923 ,,,,,,, 20.8
    1924 ,,,,,,, 20.7
    1925 ,,,,,,, 21.7
    1926 ,,,,,,, 22.2
    1927 ,,,,,,, 23.4
    1928 ,,,,,,, 26.2
    1929 ,,,,,,, 29.9
    1930 ,,,,,,, 22.4
    1931 ,,,,,,, 18.1
    1932 ,,,,,,, 14.8
    1933 ,,,,,,, 15.3
    1934 ,,,,,,, 16
    1935 ,,,,,,, 15.8
    1936 ,,,,,,, 16.2
    1937 ,,,,,,, 16.1
    1938 ,,,,,,, 14
    1939 ,,,,,,, 13.6
    1940 ,,,,,,, 12.9
    1941 ,,,,,,, 14

  38. Gravatar of Saturos Saturos
    6. July 2012 at 18:02

    “When we talk about the equation of exchange we are just talking about definitions.”

    Yes, but velocity is still a real (observable) rate of turnover of the money stock. It helps to distinguish between actual and desired velocity. Better still, think in terms of supply and demand for the stock of money.

    John, I thought velocity was displacement over time?

  39. Gravatar of ssumner ssumner
    6. July 2012 at 18:07

    Nanute, What is a negative multiplier?

    Mark, Thanks, Noah corrected his post.

    Nick, It wasn’t obvious to me either.

    Jeff, I want to target NGDP forecasts, so the data lag isn’t really an issue.

  40. Gravatar of ssumner ssumner
    6. July 2012 at 18:26

    Saturos, Velocity isn’t really the rate money turns over in transactions, it’s the ratio of NGDP to money.

  41. Gravatar of Saturos Saturos
    6. July 2012 at 18:37

    Scott, I suspect you’re getting slightly confused here. There’s no conflict between what you said and what I said. Velocity is the average number of times a dollar changes hands in a year. It is also the ratio of the money stock to NGDP. Obviously.

  42. Gravatar of Saturos Saturos
    6. July 2012 at 18:38

    Ratio of NGDP to the money stock, I should say.

  43. Gravatar of Saturos Saturos
    6. July 2012 at 18:49

    I agree with FEH. And Nick’s first comment.

  44. Gravatar of nanute nanute
    7. July 2012 at 01:12

    Scott:
    I’d say a negative multiplier is when the ratio falls below 1. See here: http://research.stlouisfed.org/fred2/series/MULT/

  45. Gravatar of Jeff Jeff
    7. July 2012 at 03:20

    Scott:

    “I want to target NGDP forecasts, so the data lag isn’t really an issue.”

    You’ve advocated NGDP futures, so data lags and the interpretive work that goes into creating the figures the futures are bets on will have to be dealt with. I’m not saying it’s impossible, only that it’s not as simple as you sometimes seem to imply it is.

    Your post was asking why no one targets NGDP, not NGDP forecasts. I was just supplying one possible reason.

    Monetarists in the sixties and seventies by and large did not trust the Fed, so targeting a forecast made by that same Fed might not have seemed like a very good idea to them.

  46. Gravatar of Jeff Graver Jeff Graver
    7. July 2012 at 05:56

    Nick: “What is the correct word for what a tachometer measures? V has the units 1/time[?]”

    Frequency.

    Going that direction implies switching to a signals processing analogy and away from the conservation of momentum analogy.

    I suspect that would spur some positive developments in the debate.

  47. Gravatar of Major_Freedom Major_Freedom
    7. July 2012 at 07:35

    Full Unemployment Hawk:

    “Velocity has no independent meaning apart from being defined as a relation between the other variables: P*Y/M.”

    That is true for realized velocity. But monetarism’s concept of velocity is desired velocity, how often people want to spend their money holdings during some period of tims, which is the reciprocal of the demand for money for a given level of P and Y (if we are talking about income velocity, rather than transactions velocity).

    That is still P*Y/M, only now it is “desired” P*Y/M.

    In equilibrium actual and desired velocity are equal, out of equilibrium they are not.

    The economy is never in equilibrium.

    While only actual velocity can be readily measured, the theoretical work on velocity is with respect to desired velocity, which Friedman, during most of his career argued was a stable function of a few variables.

    Yes, P*Y/M.

    For any given M, a change in desired velocity will cause people to spend their money more frequently, which will cause a change in P or Y or both. Friedman during most of his career argued that such changes in desired velocity are not a major factor in affecting P and Y.

    Again, there is no independent nature to velocity that has a causal effect on people’s spending. You’re fallaciously treating V in M*V=P*Y as if it were an independent variable like M, P and Y. It isn’t. V is FULLY dependent on P*Y/M.

    You have it backwards. It is the change in Y and P and M that changes people’s spending on Y, such that prices are paid, and thus V is formed. V is not an independent variable that causes spending, prices or output, through “desired velocity”.

    Your assertion above shows that you really do not have a good understanding of monetarism.

    Nice try, but your assertions above show that do not understand velocity nor the equation of exchange.

  48. Gravatar of Major_Freedom Major_Freedom
    7. July 2012 at 07:44

    Saturos:

    Yes, but velocity is still a real (observable) rate of turnover of the money stock. It helps to distinguish between actual and desired velocity. Better still, think in terms of supply and demand for the stock of money.

    Desired velocity is as fully dependent on P*Y/M as is actual velocity. It has no independent causal effect on P, Y or M. It is a function of P, Y, and M.

    You and FUH are both misinterpreting the equation of exchange. You see M*V=P*Y, and you’re treating it like a physics equation, as if each of the four variables can be “controlled” independently.

    In reality, V is not independent at all, but is determined by the other three variables.

    “Desired velocity” is actually just a “desired” P*Y/M ratio. Actual or desired V cannot possibly affect any of the variables in P*Y/M.

    I cannot for example say something like “I desire to change my actualized velocity to a new desired velocity in the future. This will have the causal effect of changing output, prices, and/or money supply.”

  49. Gravatar of ssumner ssumner
    7. July 2012 at 07:45

    Saturos, You said;

    “I suspect you’re getting slightly confused here. There’s no conflict between what you said and what I said. Velocity is the average number of times a dollar changes hands in a year. It is also the ratio of the money stock to NGDP. Obviously.”

    No it isn’t. Dollars change hands for many reasons, only a small proportion of transactions involve final goods that compose NGDP.

    nanute, OK, but most people call fractions larger than zero “positive numbers.”

    Jeff, Fair enough, in the context of the 1960s debate that makes sense.

  50. Gravatar of SecretAgent SecretAgent
    7. July 2012 at 08:00

    ssumner:

    Velocity isn’t really the rate money turns over in transactions, it’s the ratio of NGDP to money.

    In the traditional equation of exchange it isn’t. In the equation of exchange, velocity includes all transactions made in money, including financial securities and wages, rather than just the transactions made on final output as per NGDP.

    The ratio of NGDP to money is a special velocity concept, which could be written as

    V(NGDP) = P(NGDP)*Y(NGDP)/M

    In general,

    V = P*Y/M = [P(NGDP)*Y(NGDP) + P(non-NGDP)*Y(non-NGDP)]/M

    So

    V(NGDP) = V – P(non-NGDP)*Y(non-NGDP)/M

    where in a capitalist economy (that includes wage payments and perhaps financial securities):

    V(NGDP) < V

    and in a "early and rude state of society" (that contains no wage payments and only spending on final output):

    V(NGDP) = V

  51. Gravatar of nanute nanute
    7. July 2012 at 09:56

    Scott,
    Thanks. (I think.) I still haven’t gotten an answer to my first question, and now you’ve created a situation that requires my asking an additional question. In your most recent response: “OK, but most people call fractions larger than zero “positive numbers.” Is it your contention that the multiplier is not negative? Or does your quotation marks on “positive numbers”, mean otherwise.

  52. Gravatar of Jim Glass Jim Glass
    7. July 2012 at 12:05

    Conflicting definitions shouldn’t be allowed to become the root of arguments…

    “Velocity … is the ratio of NGDP to money.”

    In the traditional equation of exchange it isn’t … The ratio of NGDP to money is a special velocity concept, which could be written as …

    FRED: “Velocity is a ratio of nominal GDP to a measure of the money supply (M1 or M2). It can be thought of as the rate of turnover in the money supply — that is, the number of times one dollar is used to purchase final goods and services included in GDP.”

    That’s the default meaning of published numbers for velocity today. Alternative special concepts of velocity should be used only with disclosure and explanation of the reason why.

    ————–

    Is it your contention that the multiplier is not negative?

    FRED gives the latest measure of the M1 multiplier as 0.859. That’s a positive number.

    As the M1 multiplier is defined as the ratio of M1 to the monetary base, it is difficult to see how it could be negative.

    Subtracting the monetary base from M1 would give a negative number as a result — but that’s not the money multiplier.

    Or does your quotation marks on “positive numbers”, mean otherwise.

    Did your quotation marks in your original comment…

    Haven’t looked lately, but last time I did the multiplier was “negative.”

    … mean otherwise?

  53. Gravatar of Philip George Philip George
    7. July 2012 at 18:06

    “But here’s the mystery, almost no one in the 1950s, 1960s, or 1970s was actually advocating NGDP targeting!”

    There’s no mystery. In those decades inflation was the principal worry, so the aggregated nominal NGDP was of no use. You needed to separate inflation from GDP to get a meaningful target.

    The equation M*V=P*Y neatly encapsulates all that is wrong with modern macroeconomics. It ignores financial assets.

    The correct equation would be:
    M1*V1 + M2*V2 = P*Y + sum of (P1*F1)

    where M1 is the money used for real transactions and M2 is the money used for financial transactions. Irving Fisher thought of money in these terms and even Keynes’s General Theory had an inkling (but only a vague inkling) of this idea.

    Because they ignore the second term, economists of all shades cannot think of a situation where there can be huge monetary expansion without any inflation in the price of real goods or assets. This is exactly the situation before the Great Crash of 1929 and the period before the start of Japan’s Lost Decade.

  54. Gravatar of James in London James in London
    7. July 2012 at 23:57

    Major Freedom: Where is your own blogsite?

  55. Gravatar of nanute nanute
    8. July 2012 at 00:41

    Jim Glass,
    I assume your comment was directed at me? I merely posed an honest question early on in the thread. I made no suppositions or definitions, with the exception of the “negative” multiplier. Negative was in quotation marks to imply that the multiplier was less than a whole number. Meaning below 1. The question is still this: does the multiplier level have a correlation to the level of the velocity of money? Does M1 circulate at a higher velocity when the level is above 1 vs. when it is below 1? Care to speculate on why positive was in quotes? Scott seems to have left the building.

  56. Gravatar of Saturos Saturos
    8. July 2012 at 01:04

    Scott, Jim Glass and SecretAgent are right. There are two different velocities, V(T) and V(Y), it depends which one you use. Whether you’re using the original Equation of Exchange MV = PT or the adjusted one MV = PY, where T = vY and V(Y) = V(T)/v. That’s how my Mankiw textbook explains it. And most textbooks state the obvious fact that the ratio of whichever measure of the money stock to either total price-adjusted transactions or total newly-produced nominal final output is equivalent to looking at the average number of times that a dollar from that money stock is spent on that quantity of stuff.

    Money is fungible, you see, so it doesn’t matter if “in reality” a bunch of notes with specific serial numbers sit under my mattress forever – we can account for people’s spending of their money balances using a FIFO method, “as if” every dollar were spent after being received. Then the ratio of money balances to nominal income for each person is equal to the number of money balances each person spends that year, which equals the number of times the money stock must circulate in a year to create that nominal income. So it is true to say that velocity = average number of times a dollar is spent “buying” either T or Y.

    It’s all really quite straightforward, I swear! Ask Nick, or any other macroeconomist.

    James in London, good question.

  57. Gravatar of Saturos Saturos
    8. July 2012 at 01:06

    And yes, it’s theoretically possible to have V < 1.

  58. Gravatar of nanute nanute
    8. July 2012 at 03:25

    Saturos:
    Theoretically, V <1? I think it is there right now in reality.

  59. Gravatar of Mike Sax Mike Sax
    8. July 2012 at 03:47

    Phillip Geroge, let me just show my ignorance-what is F1?

  60. Gravatar of flow5 flow5
    8. July 2012 at 07:24

    So I have to prove you wrong? So be it. Your NgDp target analysis is flawed to begin with & it’s easy to discredit.

  61. Gravatar of Saturos Saturos
    8. July 2012 at 09:11

    nanute, what measure of the money supply are you using? Hard to believe there are more than 15 trillion dollars of any sort out there.

  62. Gravatar of Full Employment Hawk Full Employment Hawk
    8. July 2012 at 09:42

    “I cannot for example say something like “I desire to change my actualized velocity to a new desired velocity in the future.” That shows a complete misunderstanding of the concepts.

    Actual or realized velocity IS a residual equal to (P x Y)/M, and is determined by them. This is not a decsion variable for people. But desired velocity is a decision variable that individuals control.

    I, you, and everybody else, at any GIVEN level of P and Y, can decide I want to REDUCE the amount of money I have in my cookie jar or checking account. To do this I increase my spending at that level of P and Y. This increases desired velocity. Similarly I can decide I want to INCREASE the amount of money I have in my cookie jar or checking account. As a result, at any given P and Y, I reduce my spending. This reduces velocity.

    In other words people, by decreasing or increasing the money balances they have at any given level of nominal income, increase of decrease desired velocity. Both monetarism and Keynesian economics agrees with this.

    Decisions by people to decrease or increase their money balances, that is, increase or decrease their desired velocity, result in increases or decreases, respectively, in expenditures, (either directly, or indirectly working through expenditures on financial assets and therefore interest rates), and the increased/decreased expenditures increase/ decrease P and Y at any given money supply.

    Using the equation of exchange M x V = P x Y, with V being DESIRED income velocity, implies that with given M, if V goes up, P and/or Y increases to make the equation hold. Similarly, if V goes down with a given M, P and/or Y goes down.

  63. Gravatar of Major_Freedom Major_Freedom
    8. July 2012 at 12:37

    Full Unemployment Hawk:

    “I cannot for example say something like “I desire to change my actualized velocity to a new desired velocity in the future.”

    That shows a complete misunderstanding of the concepts.

    If the “concepts” cannot be interpreted in that way, then they are simply not related to economic science.

    What you are saying is not desired velocity, but a desired ratio between spending and cash balances. If that is what desired velocity means, then we already have a name for that. It’s called cash preference.

    Actual or realized velocity IS a residual equal to (P x Y)/M, and is determined by them. This is not a decsion variable for people. But desired velocity is a decision variable that individuals control.

    I, you, and everybody else, at any GIVEN level of P and Y, can decide I want to REDUCE the amount of money I have in my cookie jar or checking account. To do this I increase my spending at that level of P and Y. This increases desired velocity.

    No, you can only do so by purchasing more goods or the same goods at higher prices, because spending more money more often means there has to be more production. You cannot simply demand to spend more of your money on more goods and then poof the goods are there and thus you achieve your desired additional velocity. If the supply of goods does not increase, then a higher demand that is represented by your additional desire to spend more money, will simply raise prices, which means you cannot hold P constant.

    Similarly I can decide I want to INCREASE the amount of money I have in my cookie jar or checking account. As a result, at any given P and Y, I reduce my spending. This reduces velocity.

    The same principle applies as above, but in reverse.

    In other words people, by decreasing or increasing the money balances they have at any given level of nominal income, increase of decrease desired velocity. Both monetarism and Keynesian economics agrees with this.

    They both fail to take into account production, which is why they both treat velocity as exogenous.

    Decisions by people to decrease or increase their money balances, that is, increase or decrease their desired velocity, result in increases or decreases, respectively, in expenditures, (either directly, or indirectly working through expenditures on financial assets and therefore interest rates), and the increased/decreased expenditures increase/ decrease P and Y at any given money supply.

    Yes, THAT is something I can agree with. Now P*Y is changing.

    Using the equation of exchange M x V = P x Y, with V being DESIRED income velocity, implies that with given M, if V goes up, P and/or Y increases to make the equation hold. Similarly, if V goes down with a given M, P and/or Y goes down.

    But that is not the direction of causality. That’s the problem you are having by treating the equation like a physics equation. It is not true that a person changing their desired velocity will make Y or P increase. The direction of causality is that people desire to spend more money relative to their cash balances, at that will affect P*Y from one direction, and production will affect P*Y from the other direction.

    I cannot increase productivity by merely desiring to spend my money more often, that is, spending more quickly after each paycheck or product sales receipt. You cannot say that if desired spending increases, then Y “must” increase to “balance the equation.” Remember, the equation ONLY says money spent is equal to money received. If a person desires to spend more money, then others will of course receive more money. This doesn’t mean production goes up. Production goes up apart from velocity. Prices are in part affected apart from velocity since prices are in part a function of supply.

  64. Gravatar of Full Employment Hawk Full Employment Hawk
    8. July 2012 at 14:15

    “Full Unemployment Hawk”

    Since you have once again decided to revert to acting like a 10 year old and resort to name calling, this conversation is over.

  65. Gravatar of Full Employment Hawk Full Employment Hawk
    8. July 2012 at 14:15

    “Full Unemployment Hawk”

    Since you have once again decided to revert to acting like a 10 year old and resort to name calling, this conversation is over.

  66. Gravatar of Secret Agent Secret Agent
    8. July 2012 at 14:25

    Full Unemployment Hawk:

    Since you have once again decided to revert to acting like a 10 year old and resort to name calling, this conversation is over.

    Again, as I noted here, I call you Full Unemployment Hawk because I find your self-given name ironic. You call yourself a full employment hawk, yet I am convinced your ideas if/when practiced actually exacerbate unemployment.

  67. Gravatar of Advocates of Reason: 9 July 2012 | Economic Thought Advocates of Reason: 9 July 2012 | Economic Thought
    9. July 2012 at 00:01

    [...] Money velocity and NGDP, by Scott Sumner.  The title is misleading, because the post doesn’t really talk much about what the velocity [...]

  68. Gravatar of ssumner ssumner
    9. July 2012 at 12:15

    Secret Agent, It’s the 21st century, time to get with the new terminology. It’s also true that “gay” used to mean “cheerful.” Try using it that way today. :)

    Nanute, Yes, of course it’s positive.

    Saturos, That’s a different argument, but I’m still not buying. Lots of final goods transactions occur without money being transacted. Barter, or financial assets may be exchanged. There are different types of money, etc.

  69. Gravatar of Saturos Saturos
    10. July 2012 at 21:31

    Scott, you said,

    “There are different types of money, etc.”

    My argument is “for a given type of money”.

    “Lots of final goods transactions occur without money being transacted”

    If that happened often, then we wouldn’t call MV “nominal spending [on output]“. Not unless we violated your premise that V is the ratio of [some] M to nominal output. Furthermore, if barter was common then drops in spending wouldn’t cause such large increases in unemployment. As Nick Rowe will happily explain to you.

    More broadly, you need to accept that money matters as the medium of exchange as well as the unit of account. Because money is the unit of account, supply and demand for money determines nominal spending. And because money is the medium of exchange, nominal spending determines nominal output (the circular flow of income). (Sticky nominal wages only matter because we have to pay workers with money.)

    nanute, we can make numbers smaller by multiplying them by fractions less than 1, as well as negative numbers.

  70. Gravatar of ssumner ssumner
    11. July 2012 at 08:34

    Saturos, You said;

    “If that happened often, then we wouldn’t call MV “nominal spending [on output]“. Not unless we violated your premise that V is the ratio of [some] M to nominal output.”

    I have to disagree. Nominal spending refers to the use of dollars as the unit of account. It says nothing about whether money changed hands. If I buy a car and pay with T-bills, the transaction is a part of NGDP, but no money changed hands. The ratio of M to NGDP is V regardless of what assumptions you use about transactions technology, it’s just a definition, and hence can never be vioted by real world facts.

  71. Gravatar of Saturos Saturos
    12. July 2012 at 12:48

    Scott, I didn’t want to do this, but you leavae me no choice:

    The important magnitude, called the velocity of circulation, or rapidity of turnover, is simply the quotient obtained by dividing the total money payments for goods in the course of a year by the average amount of money in circulation by which those payments are effected. This velocity of circulation for an entire community is a sort of average of the rates of turnover of money for different persons. Each person has his own rate of turnover which he can readily calculate by dividing the amount of money he expends per year by the average amount he carries.

    Note: he’s dividing the total flow of money exchanged for goods (and not the nominal value of output) by the money stock.

    Let us denote the total circulation of money, i.e. the amount of money expended for goods in a given community during a given year, by E (expenditure); and the average amount of money in circulation in the community during the year by M (money). M will be the simple arithmetical average of the amounts of money existing at successive instants separated from each other by equal intervals of time indefinitely small. If we divide the year’s expenditures, E, by the average amount of money, M, we shall obtain what is called the average rate of turnover of money in its exchange for goods, E/M, that is, the velocity of circulation of money.*16 This velocity may be denoted by V, so that E/M = V; then E may be expressed as MV. In words: the total circulation of money in the sense of money expended is equal to the total money in circulation multiplied by its velocity of circulation or turnover.

    Also relevant is this quote:

    An algebraic statement is usually a good safeguard against loose reasoning; and loose reasoning is chiefly responsible for the suspicion under which economic theories have frequently fallen. If it is worth while in geometry to demonstrate carefully, at the start, propositions which are almost self-evident, it is a hundredfold more worth while to demonstrate with care the propositions relating to price levels, which are less self-evident; which, indeed, while confidently assumed by many, are contemptuously rejected by others.

    Of course, I’ve been quoting from Irving Fisher, “The Purchasing Power of Money”, Chapter 2, “PURCHASING POWER OF MONEY AS RELATED TO THE EQUATION OF EXCHANGE”

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