Home > Economics, Empirical Analysis > Cancelling on both sides of the equation.

Cancelling on both sides of the equation.

In the great turkey shoot that is modern economic theory, one particular formula stands out: the Equation of Exchange, sometimes also known as the Quantity theory of Money.

The quantity theories of money – it turns out there are several – attempt to formulate the idea that the total quantity of money has some role in determining the price level. The Equation of Exchange is typically stated as:


where M is the total quantity of money, V is the velocity of circulation of money, P is the price level and Q is the total quantity of aggregate transactions. The equation was  proposed in this form by the American economist Irving Fisher, in 1911 in his book The Purchasing Power of Money, its Determination and Relation to Credit, Interest and Crises. (Alternate forms use MV = PT, which replaces Q quantity of transactions with T value of aggregate transactions.)

Now this author takes no issue with the M = PQ part of the equation, but what exactly is the velocity of circulation of money and how does it affect the price level?

Velocity of circulation of money is the number of times a specific unit of money is spent in a given period of time. Quite what it means economically is a very interesting question, especially if you think about the wide range in the number of units of money that different currencies provide. In the rest of this post though, I’m going to concentrate on the role velocity of circulation plays in Fisher’s quantity theory of money, and specifically I am going to argue that it does not, and cannot influence the price level as suggested by that equation.

As a simple thought experiment, consider an economy with two economic agents, a single token of money, and a single widget.  The agents repeatedly swap the token of money for the widget. It seems quite obvious that it doesn’t matter how quickly or slowly they do this, the price of the widget (1 token of money) can’t change – there is only one token of money in the economy. Extending the example to more agents, widgets or tokens, it’s still fairly easy to see that it’s possession of widgets and tokens that determines the price level, not speed of exchange.

Engineers, it should be noted, are trained to check that the units in their equations are the same on both sides as a simple way to find errors. If the left hand side evaluates to furlongs per fortnight, and the right hand side is liters then something may have gone a little askew. In the equation of exchange, we have $/t on the left hand side (where t is time), $ and quantity of goods sold on the right. Interesting.

In Chapter 1 of his book Fisher helpfully illustrates his reasoning with an example:

Let us begin with the money side. If the number of dollars in a country is 5,000,000, and their velocity of circulation is twenty times per year, then the total amount of money changing hands (for goods) per year is 5,000,000 times twenty, or $100,000,000. This is the money side of the equation of exchange.

Since the money side of the equation is $100,000,000, the goods side must be the same. For if $100,000,000 has been spent for goods in the course of the year, then $100,000,000 worth of goods must have been sold in that year.

and then he shows a sample calculation:

$5,000,000 x 20 times a year
= 200,000,000 loaves of bread x $.10 a loaf
+ 10,000,000 tons of coal x $5 a ton
+ 30,000,000 yards of cloth x $1 a yard

i.e. $5 million in physical notes and coins were used to purchase a total of 240 million items of goods. Now there isn’t enough information in Fisher’s examples to dissect out the actual number of transactions for each type of good. However, since we know that each note or coin was used 20 times (the velocity of circulation) we also know that there must have been at least 20 transactions, and that the total number of transactions will always be some multiple of V. In fact it’s implied by the very definition of the velocity of circulation of money that there must always be at least ‘velocity of circulation of money’ transactions.

In other words, V cancels on both sides of the equation.

The assumption that appears to have gone unquestioned here is that the velocity of circulation of money can increase, independently of the number of transactions, and affect the price level. How is it going to do that, in the real life that is to so many economists it seems, a special case?

There is I think, an implicit assumption hidden in Fisher’s work, that is substantially incorrect. That in some sense, there are only two sides of the equation, rather than a time series. Consider what we’re told – each token of money, each dollar bill or coin, is used on average 20 times. So it’s not the case that $100,000,000 is exchanged for some amount of goods in a straight swap as the accounting relationship represents. Rather, $5,000,000 is exchanged for some amount of loaves of bread, the bread maker then presumably exchanges those notes for some amount of coal, and the coal maker goes and buys cloth. Then the cloth makers go and buy the bread.

Or in other words, again going from the original example:

$5,000,000 V
x 4 4 x 50,000,000 loaves of bread x $.10 a loaf
x 10 10 x 1,000,000 tons of coal x $5 a ton
x 6 6 x 5,000,000 yards of cloth x $1 a yard

The other questionable assumption is that each item is only traded once. If the coal miners start selling on their cloth to the bread makers, then velocity will increase, but then so does the number of transactions, and so V cancels out again.

For a more empirical disproof, consider high frequency stock trading. The volume of shares sold, and the number of associated transactions has increased enormously over the last 20 years with computer automation and algorithmic trading. Now share valuations have increased – broadly in line with the quantitative increase in the respective money supplies, but they haven’t increased by anything close to the amount that Fisher’s equation would imply if the velocity of circulation could increase the price level. Leaving aside the fascinating question of how share prices would actually behave once hedge funds discovered they could dramatically affect the price by varying the frequency of trading.

So why did Fisher get it so badly wrong?

That turns out in and of itself to be quite interesting. Fisher wrote this in 1911, during the gold standard regime, and a large part of the book is in fact dedicated to explaining the banking system, and how bank deposits interact with the circulation of money, and gold. There’s even a chapter on bimetallism. One thing Fisher it at pains to point out is that bank deposits are not money:

But while a bank deposit transferable by check is included as circulating media, it is not money. A bank note, on the other hand, is both circulating medium and money. Between these two lies the final line of distinction between what is money and what is not. True, the line is delicately drawn, especially when we come to such checks as cashier’s checks or certified checks, for the latter are almost identical with bank notes. Each is a demand liability on a bank, and each confers on the holder the right to draw money. Yet while a note is generally acceptable in exchange, a check is specially acceptable only, i.e. only by the consent of the payee. Real money rights are what a payee accepts without question, because he is induced to do so either by “legal tender” laws or by a well-established custom.

The Purchasing Power of Money, Chapter 2, Section 1

Whatever the legal situation may be though, in 1911 in terms of their influence on the price level, bank deposits were most assuredly functioning as money.  Although Fisher does later extend his formula to include bank deposits (and a separate velocity of circulation), one other thing also becomes clear as he does so – he doesn’t really acknowledge the expansion of bank deposits  caused by lending.

Of course he doesn’t, it’s 1911 – it won’t be even partially explained for another 20 years. Economists are only just starting to explore the problem of what’s causing bank deposit expansion in a supposedly stable financial world of physical note regulation by the gold standard, and Keynes’ explanation of re-deposit expansion in the Macmillan Report won’t come out until 1931.  What Fisher has actually stumbled over, without apparently realizing it, is the need at that time for some kind of fudge factor to explain the independent expansion of bank deposits relative to the quantities of physical notes and coin, and gold that are supposedly regulating them. Velocity of circulation, almost impossible to measure accurately, and easy to find random explanations to adjust when needed, is a very convenient variable, and with Economics still in complete disarray when it comes to properly understanding how bank deposit expansion behaves as an economic force, it continues to be so.

It’s a little hard to know where to begin, in terms of illustrating the continued convenience of this non-existent effect, so with apologies for picking on the authors concerned, today’s prize goes to the Bank of England’s First Quarter report for 2011, which includes an article by Bridges, Rossiter and Thomas on “Understanding the recent weakness in Broad Money growth.”

The growth rate of nominal spending (nominal GDP) has picked up sharply over the past year, despite subdued money growth (Chart 2). This means that money has had to circulate at a greater rate in the economy to finance the higher value of transactions — in other words there has been an increase in the velocity of circulation of broad money.(5) That is in contrast to the long-run downward trend observed in velocity since the 1980s.

The incremental information in broad money growth about future nominal spending has to be conditioned on a view of the outlook for the velocity of circulation. Currently broad money growth is weak, which might signal a downside risk to future nominal spending and, ultimately, inflation. But there may be reasons why both the supply of, and demand for, broad money may have changed relative to spending. That would lead to a change in the equilibrium level of the velocity of circulation. So understanding the recent factors influencing velocity and the extent to which they might persist will be important for assessing future inflationary pressures.

Bank of England Quarterly Report 2011 Q1, p22

Broad Money appears to be the M4 measurement, which as noted previously, is a horrible mixture of money in the form of bank accounts, and various kinds of debt. The “weakness” in broad money growth appears to be a drop from an annual increase of 10% or so during the credit boom, to a mere 2-3% in the last few years. Perhaps the most alarming sentence in the report, which is quite entertaining to read from the perspective that velocity of circulation of money is actually irrelevant in terms of effecting the price level, is the last one:

Should money growth continue to remain weak, then analysing the causes of this, using the types of analysis employed in this article, will be important in judging whether that weakness is signalling weak nominal spending growth in the future.

Bank of England Quarterly Report 2011 Q1, p34

Are Economists at the Bank of England actually advocating money supply growth as a positive economic indicator? If they really do believe that, then it only remains to pass on the solution described by the great Douglas Adams.

Since we decided a few weeks ago to adopt leaves as legal tender, we have, of course all become immensely rich…  But, we have also run into a small inflation problem on account of the high level of leaf availability. Which means that I gather the current going rate has something like three major deciduous forests buying one ship’s peanut. So, um, in order to obviate this problem and effectively revalue the leaf, we are about to embark on an extensive defoliation campaign, and um, burn down all the forests. I think that’s a sensible move don’t you?

Hitchiker’s Guide to the Galaxy, Douglas Adams.

Central Bankers to the B Ark please. Don’t mind the gap.
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  1. July 24th, 2011 at 03:13 | #1

    Very nice article.

    By coincidence – I’m just preparing a blog entry on money flows for the PositiveMoney.org.uk website. It talks about MV=PT. I think this equation comes about from a too simplistic model in which all purchases are of consumer goods that are manufactured, sold once, then consumed. There is then a limit on the frequency of trading determined by the time taken to manufacture each item. I put a temporary mock-up of my blog entry here:


    It looks to me like most economic models assume that most purchases are goods to be consumed rather than traded. But in modern times, a greater and greater fraction of things purchased with money (by anyone, including bankers) are being swapped back and forth and scarcely consumed at all. For example shares, gold and land. This fact renders many economic models completely broken.

  2. Angry Saver
    August 3rd, 2011 at 14:43 | #2


    I’m a big fan of the Hitchhiker series. The concept of a fourth book in a trilogy never left me. As for the quantity theory and velocity, the whole idea doesn’t sit right with me.

    And as an engineer, I can attest that engineers regularly check the units on both sides of an equation. What about all the trades that take place on Wall St. each and every day? HFT, etc. Those aren’t even measured.

    Eagerness and vanity play large economic roles imo, yet nobody ever tries to quantify those.

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