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Monday, November 1, 2021

Why We Cannot Measure Money Velocity Directly

Note: this is an unedited draft of a section from my inflation book manuscript. I wanted to avoid theoretical wrangling within my book, so I am not entirely happy about adding it. That said, money velocity shows up so often in the context of inflation that I feel that we need to cover factual statements about its determination.

Given the importance that many people attach to the velocity of money, I feel that I need to offer a longer discussion of the weakness of the concept. Since it is somewhat of a distraction from the flow of this text, I have put it into this appendix section that can be readily skipped if the reader is blissfully unaware of money velocity.

I explain the problems by first starting with a very simple idealised economy in which money velocity is well-defined and is perhaps somewhat useful. I then add complications that appear in the real world which make velocity effectively impossible to measure.

When Velocity Works

Let us imagine a society where all transactions take place with $10 bills. We then assume that everyone in this society holds a $10 bill for one hour before going out to spend it. We see that for each $10 bill in that society, it will trigger 24 transactions in a day, with a total value of $240. If we look at the transaction version of the equation of exchange:

MV = PT,

where M = money supply, V = velocity, P = average price of transactions, T is the volume of transactions, then V=24. (Note that I typically use MV=PQ, where Q is the quantity of transactions in real GDP, but we are now looking at all transactions. Since there are transactions that will have no effect on GDP, we see that PT is a wider concept.)

What happens if everyone decides to hold their $10 bills for two hours? The “velocity” of transactions falls, and each $10 bill only triggers $120 in transactions per day. This gives us a reason to think “velocity” is useful – it looks like the inverse of the holding time. Meanwhile, if we assume if velocity is constant, we can then see that if we add another $10 bill to the economy, we add $240 in daily nominal transactions. It appears that money creation can determine aggregate demand.

Unfortunately, things are more complicated.

Need to Cover all Monetary Assets

We can make a very simple addition to the model: $100 bills are added to the mix. However, people hold these bills longer – for four hours on average.

This means that each $100 bill turns over only 4 times in a day, so it adds $400 to daily transactions.

We then see that if we omitted the $100 bills from our money supply measure (M), we are missing transactions. If we have 10 $10 bills and 2 $100 bills, a version of M that ignores the $100 bills gives us daily transactions of $2400 (=10×$240), while the actual level of transactions would be $3,200 (=10×$2400 + 2×$400).

If we properly measure M as being $300 (=10×$10 + 2×$100), we get a velocity of 10.67, which is an average of the velocities of the two types of bills.

Measurement Requires Tracking All Monetary Holdings

In a fiat currency, issuers normally keep track of all the monetary instruments that they have issued. (If we use a commodity currency like gold, we could have private sector entities produce things like lumps of gold that might act like money, and this is unlikely to be tracked.) We use these totals to define the money stock. However, we run into problems if we want to calculate velocity.

If we do not track the monetary instruments as they are used in transactions (which could happen with electronic money), all we can do is survey entities to get an idea what their money holding period is. This poses many practical difficulties.

  • It appears unlikely that many entities track their money holding periods.

  • The survey needs to be aligned to monetary holdings – we would need to weight responses by the amount of money they respondent holds.

  • Unless we do a complete census of monetary holdings, we do not know how many monetary instruments are either lost or circulating outside the domestic economy.

  • We can have counterfeit money injected into the system.

The result is that we cannot hope to capture velocity for banknotes and coins, which is a core part of the money supply. We would need an electronic version of money where all transactions can be monitored by state statistical agencies.

Money Is Created and Destroyed

The next stumbling block is that monetary instruments are both created and destroyed. For example, if banknotes are the only form of money – as in the previous examples – then the issuer of the money will typically issue it as part of a transaction. That is, it will create money and register a transaction coincident with the money creation. What is the “holding time” for money that has popped into existence just when a transaction takes place? Meanwhile, money will also be destroyed via having it returned to the issuer.

We end up needing to create some average of the amount of money during the accounting period – which is typically not how monetary aggregates are measured in practice. Instead, they are measured as of the end of some accounting day.

Although this may not appear to be a serious problem for banknotes and coins, it is a concern once we widen our definition of “money” to include credit instruments like bank deposits. Bank deposits are created and destroyed in aggregate mainly by the extension and paying back of bank loans respectively.

Do Not Need to Use Money for Transactions

In real world monetary economies, we do not need to use instruments that appear in monetary aggregates in order to have a transaction. The value of the transaction is measured in terms of the monetary unit, but we do not need to use “money.”

  • Households routinely pay for goods and services using credit cards.

  • Businesses routinely pay for goods and services on account: they have an increase in their accounts payable, the vendor has an increase in their accounts receivable.

  • Some businesses (mainly start ups) will sometimes issue equity to pay bills.

  • Barter transactions are undertaken – a reciprocal swap of goods and services between two entities. If the entities do not attempt to avoid taxes, governments would typically treat the barter deal as a pair of monetary transactions of equal value and assess taxes accordingly.

The first two cases – using credit to make a purchase – is the most common, and no money changes hands. One can attempt to salvage the role of money and argue that those credit instruments will be settled with money. The problem with that argument is that this is a separate transaction, and might occur in a different accounting period. Why are attempting to include a future transaction in an relationship that is supposed to hold within an accounting period?

Not All Transactions Show Up In GDP

The usual reason why we are allegedly interested in the MV=PQ relationship is that we take Q to be real GDP, P to be the GDP deflator, and so PQ is nominal GDP. However, our holding period model just tells us that there are transactions – not whether those transactions add to GDP. The only transactions that add to GDP are those related to the purchase of goods and services currently produced (or in inventory), and not financial transactions or secondary trading of existing real assets.

For example, the purchases of equities on the stock market does not add to GDP (excluding fee payments, which are income for financial intermediaries). This means that GDP is not affected by an increased pace of trading on the stock exchange if investors decide to turn over their portfolios at a quicker pace. However, all those trades would show up as transactions if we were able to track every single monetary transaction within the economy.

This means that we cannot relate “money holding times” to the measured velocity V that results in MV=PQ. We need to calculate PQ, and then back out V via the relation V = PQ /M.

It is possible that someone might be interested in looking at all transactions, or MV = PT. The problem with that specification is straightforward: how do we calculate P? We would need a “price index” that somehow corresponds to all transactions in the economy. What is the “price” of the transaction “loan me $20 until Friday?” Trying to assign a price index to every single traded financial asset is not a plausible exercise, nor is it clear that even if it were attempted, the result would make any sense.

Why Are We Looking at V?

If we are attempting to compare money growth to nominal GDP, the discussion above explains that why we cannot hope to independently estimate V based on some other factors, V can only be computed via its definition: V = PQ/M.

Is the level of V in any sense meaningful? Not really, since it will depend upon which of the many potential definitions you choose for M. Is the change in V meaningful? Not really. If V rises (falls), it is just a way of saying that the growth rate of nominal GDP was higher (lower) than the growth rate of a chosen M. We get that information from just looking at the growth rates themselves.

The only reason to care about V is if we believe that it can be predicted, and there is no sign that anyone can do this – other than predicting both PQ and M, and backing out the change. But if we can predict PQ already, why do we care about V?

My feeling – which many might disagree with – is that velocity adds value if you believe that it “ought” to be constant. Many people follow the primitive Quantity Theory of Money and assume that velocity is indeed constant. However, there is no reason to believe the primitive Quantity Theory of Money, and so there should be no reason to invoke velocity in the first place.

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(c) Brian Romanchuk 2021


  1. 《Meanwhile, money will also be destroyed via having it returned to the issuer.》

    Why can't the issuer reinvest, thus keeping the money in circulation? Does the Fed reinvest maturing bond proceeds in new purchases, thus choosing not to destroy money as loans are repaid? Can private banks similarly keep money from paid-off loans in circulation, by creating deposits for some other borrower?

    1. The money is removed from circulation/destroyed. The issuer needs to undertake another transaction to create more, but that is not automatic.
      - Maturing bonds at the central bank just moves around intra-governmental accounts, and has no effect on the money supply.
      - Banks need to go through a lending process. That lending process is slow, and cannot be fine tuned, other than for an extremely small bank.

  2. This comment has been removed by a blog administrator.

  3. 《Maturing bonds at the central bank just moves around intra-governmental accounts, and has no effect on the money supply.》

    Why did the Fed end its money destruction in 2019, before Covid was ever mentioned?

    Why can't a bank immediately and automatically reinvest maturing loans, as the Fed has done since QE started, except for the brief attempt to destroy money in 2019 which ended because it precipitated a repo crisis?

    《Maturing bonds at the central bank just moves around intra-governmental accounts, and has no effect on the money supply.》

    Why don't the reserves created out of thin air end up being balanced by bank money deposits in MMFs and other retail deposit accounts? If you sell toxic assets to the Fed, don't your liabilities decrease, so you can balance the inflow of reserves by increasing equity?

    1. 1) You’d need to read the news stories. Presumably, the Fedthought they had policy at the correct balance.
      2) Banks cannot force other entities to take out loans,

      Later points: in every market exchange, entities trade assets with other entities. The asset exchanges always balance (other than mark-to-market gains/losses). Liabilities are not affected.


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