Velocity Matters

An accounting identity does not indicate the direction of causation. Not ever.

I’ve been caught out on this a few times myself, usually when I am trying to deduce something useful from national accounting equations. But I’m merely a writer. People actually involved in the formulation of policy should know better.

Here’s an attempt by people who should know better to try to infer the direction of causation from an identity. On the St. Louis Federal Reserve’s blog site is this post by Yi Wen and Maria Arias. It purports to show that the reason why three rounds of QE in the US have failed to raise inflation is because the velocity of money has collapsed. And they then come up with some reasons why velocity has collapsed, though sadly no ideas about what to do about it.

Their argument is based on the familiar Quantity Theory of Money:


where M is the monetary base M0, P is the price level, Y is real GDP, and V is the velocity of money. Sometimes real economic output Q is used as a substitute for Y.

What this equation shows is that NGDP is determined by the quantity of money in circulation and the velocity of transactions in the economy. Or – does it? Written like this, it actually suggests that the quantity of money in circulation and its velocity are determined by NGDP, not the other way round, since the identity on the left hand side is the dependent variable. But this is an identity. It could just as easily be written PY = MV. Indeed, sometimes it is.

But this equation does suggest that an increase in the monetary base should be associated with rising NGDP. The monetary base has increased dramatically since 2008, as this chart shows:

Wow. If this equation is right, then there should be a simply massive increase in either inflation or GDP or both. Here’s what the writers say:
Based on this equation, holding the money velocity constant, if the money supply (M) increases at a faster rate than real economic output (Q), the price level (P) must increase to make up the difference. According to this view, inflation in the U.S. should have been about 31 percent per year between 2008 and 2013, when the money supply grew at an average pace of 33 percent per year and output grew at an average pace just below 2 percent. Why, then, has inflation remained persistently low (below 2 percent) during this period?
And they go on to conclude that V must have collapsed. So far so good. Mathematically, this must indeed be the case, since

V = PY/M

and we know that the numerator has not increased to match the rise in the denominator. But then their analysis goes horribly wrong.

They say that V falls when there are fewer transactions in the economy. This is true, but it is not the only possible explanation for such a large fall. The increase in the monetary base itself is sufficient to cause V to collapse even if the number of transactions in the economy does not change. This is because when there is more monetary base it does not need to “churn” so quickly in order to support the same number of transactions. It is simple mathematics.

The writers correctly describe the sharp decline in velocity as “offsetting” the sharp increase in money supply, but they then claim that this “leads to” almost no change in output and inflation. No it doesn’t. The causation could be entirely the other way round: stagnant output and low inflation cause velocity to fall when the monetary base is increased. We simply do not know which side of the equation is causative. Never, ever, try to deduce causation from an identity.

However, the writers then redeem themselves by asking the right question:

Why did the monetary base increase not cause a proportionate increase in either the general price level or GDP?
 Why indeed. The answer, apparently, lies in this chart:

The writers claim this chart shows that people have been hoarding money instead of spending it. The fall in velocity is due to “an unprecedented increase in money demand”, caused by very low interest rates which have encouraged people to switch from interest-bearing assets into money.

But this chart doesn’t show anything of the kind. What it shows is the rise in reserves due to QE: the three rounds of QE can be clearly seen. The sharp-eyed among you will notice that the reserve balances in this chart and the adjusted monetary base in Chart1 look very similar. That is because they are almost the same thing. The adjusted monetary base is largely made up of bank reserves.

The quantity of reserves in the monetary system is determined by the Federal Reserve and cannot be changed by commercial banks. Lending does not reduce reserves. Nor does spending. And hoarding doesn’t increase them, whatever the writers of this piece may think.

Commercial bank customers can reduce reserves by converting them to physical currency. But if demand for physical currency were rising, currency in circulation should show a deviation from its pre-2008 trend. No such deviation is visible.

The truth is that the enormous increase in reserve balances is entirely due to asset purchases by the Federal Reserve. This chart tells us absolutely nothing about what people have been doing with their money. It cannot be used to deduce that people are hoarding money instead of spending it.

So the writers made two errors: they inferred causation from an identity, and they misunderstood the cause of high reserve levels in the banking system. And because of these errors, they then go on to draw an entirely unwarranted and very dangerous conclusion:

In this regard, the unconventional monetary policy has reinforced the recession by stimulating the private sector’s money demand through pursuing an excessively low interest rate policy (i.e., the zero-interest rate policy).
There is a further error here, of course: the zero-interest rate policy is by no means solely due to unconventional monetary policy. But leaving that aside, it simply is not possible to deduce from the evidence presented in this piece that very low interest rates have made the recession worse.

QE does, of course, encourage people to substitute cash for bonds. That is its purpose. So even though reserve balances tell us nothing at all about private sector money demand, we would expect that money demand might rise because of QE. But that does not mean that people then sit on that cash. On the contrary, the behaviour of the stock market, bond markets and commodity markets over the last few years has shown only too clearly what people do with the cash they receive in return for bonds sold to the Fed. During the era of QE, stock market reached all-time highs, junk bond yields were on the floor and commodities have shown spikes and volatility. In 2014, when the Fed ended QE, the market price distortions caused by QE unravelled: commodity prices collapsed, there was a massive sell-off in emerging market bonds and the stock market wobbled (though it has since recovered). This is hardly compatible with the writers’ assertion that people hoarded cash instead of investing in riskier assets. Indeed, “reach for yield” by investors desperately looking for better returns in a low-interest-rate environment has been a matter of considerable concern to the FOMC. If everyone was sitting on cash as these writers suppose, “reach for yield” would not be a problem, would it?

When people don’t understand how the monetary system works they make gross errors which can lead to unfortunate conclusions and damaging policy recommendations. Raising interest rates because of a mistaken belief that high reserve balances arise from people hoarding cash would be catastrophic in a recessionary economy. I find it very worrying that two Federal Reserve economists are apparently so ignorant of the system of which they are part.

But of course, they DID ask the right question. Why hasn’t such a large increase in M resulted in sharply rising output and inflation?

They are not looking at the right M. The monetary measure they should be looking at is the money actually used for transactions in the economy, which is broad money M2. And the picture for M2 is very different:


Well, well. Seems M2 growth is, er, pretty normal. M2 is increased by bank lending……so raising M0 to the skies did not have any significant effect on bank lending. The so-called money multiplier, which is the arithmetic relationship between M0 and M2, is broken. If it ever was a useful predictor of bank lending growth (which is questionable), it certainly isn’t now.

To complete the picture, this is M2 velocity:

Now, we have normal M2 growth, so - unlike the M0 velocity fall - this is genuinely a fall in velocity due to lower transaction volumes. And it is associated with stagnant NGDP.

But I’m not about to tell you which causes which. After all, one should never deduce causation from an identity.

Related reading: Arithmetic for Austrians

This post was written in 2014 and originally posted on the online magazine Pieria under the title "When Wonks Get Things Wrong". As Pieria has now folded, I have re-posted it here, updated in the light of subsequent developments.

Image from Wikipedia.


  1. And why not to think that money has gone to asset markets, mainly stocks?

    1. Read the post again. Base money cannot "go" anywhere, it is always held at the central bank accounts. And (M2) money circulates, it cannot go to any markets.

    2. Jussi I advise you read the post again. Frances explicitly makes the point that Q.E has been a boon for stock prices because of falls in long term interest rates. Also base money always sits in an account at the central bank, but it can change hands, and therefore impact asset prices. If you don't want to hold your new base money from the CB, you go and buy something with it.

    3. Chris, are you claiming that "money has gone to asset markets, mainly stocks" and therefore the velocity is down and general price level hasn't gone up. Or what's your point?

    4. I am not saying anything about asset markets, the general price level or velocity. You stated that "base money cannot go anywhere" and I am pointing out that it can and it does. Otherwise why would any central bank do Q.E?

      You also say "(M2) money circulates, it cannot go to any markets" and i'm confused by what you mean.

      Maybe I completely misunderstand your comment above but it all just appears confused.

      My point about asset prices was that Q.E has impacted asset prices but because yields have fallen not because of an increase in the money supply.

    5. Base money consists of currency held with the central bank in the reserve accounts of the commercial banks. Only commercial banks hold reserve accounts, and therefore this currency cannot ever be transferred to a non-bank. It cannot be lent or spent in the wider economy. It is used for the following:-

      (a) To settle interbank transactions. If a bank transfers funds to another bank, that is settled by transferring the relevant amount of base money from one to another.

      (b) Purchases of treasury bonds are settled by deducting the relevant bank's reserve account at the central bank. That is effectively just a transfer of a deposit with the central bank to one with the treasury. (Similarly, at maturity of a treasury bond, redemption is made by transferring the currency back to the central bank - that is all that "repayment" implies - it is a change of account held at state institutions, the currency itself does not mature.)

      (c) For payment of taxes, whether for the bank's own account or on behalf of a customer.

      (d) To purchase cash notes from the central bank on behalf of clients (who sacrifice an equal amount of their own deposits with the relevant central bank).

      And that's all. Why do central banks do QE? Simply to raise the price and lower the yield on longer term bonds issued by the treasury. And to fool people into thinking that they are injecting money into the economy or stimulating bank lending.

      QE just exchanges a deposit of currency held at the treasury evidenced by a bond for a deposit in a reserve account at the central bank. That is change of account with the state, and no new currency is created.

    6. Thank you for reciting the textbook. I am aware of everything you just said. You still miss my point entirely. Maybe I am not making myself clear enough.

      Reserve balances may never be able to leave the central bank but the point I am making is that the reserves change hands to settle transactions between people buying and selling assets in the portfolio rebalancing that occurs after a central bank announces and then engages in quantitative easing.

    7. Sorry for that. Then I agree, but also assert that the reserves do not facilitate, finance or in any other way encourage the non-bank private sector to exchange assets.

      They simply settle any interbank transfers associated with them. Banks are never short of the reserves they need to settle interbank transfers until they are close to failure. They can create the reserves in an instant by buying a treasury bond from a client and selling it to the central bank.

    8. The fault was mine! I didn't clearly state my position the first time round. I agree the level of reserves per se doesn't influence private sector decisions but Q.E does.

      Portfolio rebalancing is a thing - ask any pension fund manager.

    9. Chris, not sure what you mean by this:

      "I am not saying anything about asset markets, the general price level or velocity. You stated that "base money cannot go anywhere" and I am pointing out that it can and it does. Otherwise why would any central bank do Q.E?".

      So where does base money go? Central banks target inflation (as general price level), that's why. They have resorted to QE because interest rates were on their zero lower bound (google).

      And it is utterly confusing that you keep commenting this post ("Velocity Matters") without "saying anything about asset markets, the general price level or velocity". And yet you consistently keep bringing up QE, which is all about asset markets, the general price level and velocity.

    10. Base money doesn't "go" anywhere. It sits on deposit at the central bank, gathering dust, can be transferred to an account at the relevant treasury (gathering dust) or is used to pay taxes (in which case, it is dust).

      M2 may certainly be used to drive market purchases. If you own a deposit you can exchange it with me to buy my Apple shares or BTC. And then we can reverse that trade, and keep on going.

      Absent any taxation, that won't reduce M (it's a transfer of ownership of the relevant deposit) but does affect V, without contributing to what is generally referred to as "consumer inflation".

      The equation is nuts! And that's before you allow for the fact that M only measures the currency held on deposit at monetary institutions. In the UK, that ignores the £1,200bn held on deposit at HM Treasury in gilts, which can be liquidated and spent in an instant.

  2. Surely velocity is just the number that makes the LHS equal the RHS, since it cannot be directly measured and is simply inferred from the identity. The identity is intuitive, but useless in any epistemological sense.

    You might as well replace MV with K. You can't measure K either, but it has the advantage of simplicity.


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