Friday, 7 February 2014

Velocity and Negative Money

Nick Rowe has a post on negative money.  It makes for interesting reading, I think, because dealing with this idea must be quite a headache for economists that like to think in terms of the quantity theory of money.

Rowe focuses on money as medium of exchange so he looks to checking account balances as a measure of money.  It is therefore fairly easy for us to see negative money in the real world - it is simply the balance of accounts that are overdrawn.  We might also like to include credit card balances as negative money.  There are good reasons for doing so, although some people prefer to focus on the settlement of the credit card debt as the monetary exchange.

If we want to acknowledge the role of negative money as a medium of exchange, it raises some difficult questions when we want to consider the velocity of circulation.  Should we ignore negative balances and calculate velocity only by reference to the sum of positive balances?  Should we subtract the negative balances?  Or maybe add them?

Underlying this is a problem about what velocity of circulation means when we take into account negative money.  Consider four possibilities for a case of A buying goods from B.

1.  A and B both have positive checking account balances.

2. Both A and B are overdrawn.

3. A is overdrawn, B has a positive balance.

4. A has a positive balance, B is overdrawn.

Situation 1 is the easiest to understand in terms of velocity.  Money is being transferred from A to B.  If all payments were made this way, then in theory we could label every dollar and count how many times it changed hands to calculate the individual velocity of each.  This would relate in a natural way to aggregate velocity.

In situation 2, the flow seems to be the other way.  A's negative balance increases and B's decreases, so it seems like maybe a negative balance has passed from B to A.  Again, maybe if all payments were made this way, we could make some sense of velocity by labelling negative dollars.  But if it's happening alongside the sort of payments in situation 1, it's more problematic.  We seem to have two separate moneys circulating, but in different directions.  Subtracting one from the other doesn't make sense.  To get some kind of meaningful measure we would have to aggregate positive and negative dollars as if they were alternative currencies.  This seems to suggest that we need to add aggregate positive and negative balances to get a measure of the money supply.

Situations 3 and 4 complicate things even further.  Now it is no longer clear whether a positive dollar or a negative dollar is changing hands.  We also have the problem that the actual quantity of money is changing in the process of payment.  In case 3, the payment increases both positive and negative balances; in situation 4 it decreases both.

So not only do we not have a clear idea of what actually might be circulating; but we also have an ambiguity over how much money might be in existence at the time of payment.

One way around this takes us back to the question of credit card balances.  When I walk into a shop, the actual immediate spending power at my disposal is made up of a number of items:

1. the cash in my pocket,

2. the balance of my checking account, plus my overdraft limit,

3. the unutilised credit limit on my credit card.

Maybe our measure of money should therefore be the sum of these things.  We can see this more easily by interpreting things slightly differently.  So, for example, we view an unutilised credit card limit as being a notional loan and an equal notional positive balance on some transaction account.  To find our money supply, we then add the notional positive balances to the real positive balances.  Now all transactions involve the transfer of positive balances, although sometimes these are real and sometimes notional.

This, I think, should allow us to calculate some kind of velocity measure.  Whether it is useful or not is another matter


  1. Morris Copeland also had to say something like this at various points in his book "A Study of Moneyflows in the United States"

    1. Interesting. Thanks for the reference. I'll have a look.

    2. Yeah although he doesn't talk of negative money.

    3. NBER has made it available here Check equation of exchange in the index. In particular page 10 (Chapter 2) has some discussion but also other places. Haven't finished reading the book yet.

    4. Another good discussion in the book is how the hydraulic analogy which most economists tend to prefer is quite misleading and the circuit analogy is much better.

  2. This sort of nonsense about "negative money" just isn't necessary.

    Can't we just look at how banking works and describe it the way its described by accounting and standard operational terminology?

    1. Are "medium of exchange" and "medium/unit of account" part of standard operational terminology? If not, then no, we can't do that.

      The whole point of my post is that a lot of standard terminology, like the distinction between "cash" and "credit", is extremely misleading.

    2. Ultimately, I think we want to look at the way things are and try to draw some conclusions that will allow us to make informed decisions, as policy makers or as private agents. If using concepts like "medium of exchange" help us with that, then that's fine. But if we're struggling to fit the concept to the real world, then I'd say it's better to forget that concept and find another way of thinking about it.

      Maybe the question of what is the medium of exchange is interesting philosophically, but we shouldn't need to be rely on having an answer to that in order to be able to address questions about how the real world operates.

      The topic of this post was velocity. Velocity might be a useful concept in a world where all money takes the form of gold coins. But, in the world we live in now, its meaning is more obscure. In my view, trying to understand what it means is a waste of time.
      btw Nick - my post was not intended to counter anything in your own post - it merely prompted me to put down something I've been meaning to write for some time.

    3. Nick: " post was not intended to counter anything in your own post - it merely prompted me to put down something I've been meaning to write for some time."

      Yep. Understood.

      I really need the concepts of "medium of exchange" and "medium of account" in order to do macroeconomics. If people did not use a medium of exchange and could easily do barter, or if prices were not sticky in terms of some medium of account, then macro would be very very different. We would not observe recessions like the ones we do observe. The unemployed would immediately barter their way back to full employment. Monetary policy would not exist.

      One of the key features of a medium of exchange is that it circulates, whenever we buy or sell anything else. I can imagine an economy where people hold land and houses and bonds forever. An economy where people held the medium of exchange forever would be a contradiction. If the medium of exchange stops circulating (if velocity drops to zero) all trade stops.

      Medium of exchange, medium of account, and velocity, are concepts that are central to understanding the real world of the trade cycle (business cycle).

    4. I think we are in agreement on the idea that the recessions we get are a function of being a monetary economy rather than a barter one. The story you tell about recessions and excess demand for money is very similar to how I would describe it. The difference between us, I think, is that I would tend to look at the impact of financial assets generally, when you would look at the medium of exchange. Things like an asset's liquidity certainly matter to me, but I feel no need to draw a line between those that might be medium of exchange and those that are not.

      Also, I just don't think of anything circulating when I think of payments. If I (banking at RBC) pay you (banking at TD), my account at RBC is debited and your account at TD is credited, together with whatever arrangements RBC and TD undertake to settle. Lumping balances at TD and balances at RBC together to say that something has circulated seems to me an unnecessary fiction.

    5. "Are "medium of exchange" and "medium/unit of account" part of standard operational terminology? If not, then no, we can't do that."

      They're not - but we certainly can.

      The reason being that they're (optionally) complementary to standard terminology. There's no way you can get into serious trouble by applying them to actual banking operations. It may improve understanding. I think it probably does if applied consistently. No problem.

      But "negative money" is entirely differently. It conflicts with standard terminology.

  3. Nick: "Situations 3 and 4 complicate things even further. Now it is no longer clear whether a positive dollar or a negative dollar is changing hands. We also have the problem that the actual quantity of money is changing in the process of payment. In case 3, the payment increases both positive and negative balances; in situation 4 it decreases both."

    That seems to me to be an interesting fact.

    In a stationary economy, 3 and 4 would be happening equally often, and the money supply (green notes plus red notes) would be staying constant over time..

    In an economy where 3 was happening more often than 4, the money supply would be expanding.

    In an economy that was "deleveraging" so 4 was happening more often than 3, the money supply would be contracting.

    1. Yes, and there is a further point which I have made elsewhere but did not mention in this post. Unless settlement is carried out under RTGS, there is sometimes no way to tell which order transactions take place in. So if you have type 3 and 4 transactions happening, there may be no fact-of-the-matter as to what the actual money supply is at any particular time, other than overnight. It might rise then fall or fall then rise, but there is no objective way of deciding which has really happened.

  4. Nick E.,

    I view your post as a real world application of the idea of negative money, which I think is roughly how you intended it.

    In the empirical real world, gross positive money balances obviously exceed gross negative balances. This is evident by considering the nature of bank balance sheets and the prominence of loans and deposits etc.

    It is also the case that aggregate gross negative balances outstanding must be offset by an equal amount of aggregate gross positive balances outstanding. In a real, mixed world (with both loans and negative deposit balances), those latter positive balances in aggregate are a subset of the totality of gross positive balances.

    You can see this by first considering an inside money model in which there are no nominal loans on balance sheet at all. Then gross negative balances equal gross positive balances and net balances equal zero. No loans. No net deposits. That can be proven through a regression type argument – going back to the first payment that generates a gross negative balance and that it must therefore generate a gross positive balance, etc. From there, consider all permutations and combinations of subsequent flows, and it must be the case that gross negative balances are matched somewhere by gross positive balances. That’s in the pure model where there are no nominal loans on balance sheet and only gross negative balances instead.

    So – any gross negative balance is duplicated as a gross positive balance copy.

    In a pure world of negative balances and no nominal loans, velocity can be considered as the turnover of either gross positive or gross negative balances because the two are always equal – allowing for the fact that the stock of money is always changing. That that the money stock may change as measured either way is neither here nor there – it is also changes in the conventional loan and deposit world, because money expands and contracts along with the asset side of the bank balance sheet – loan advances and repayments etc. (as well as with liability/equity substitutions, etc.) In a pure negative money world without loans, it is changing because both negative and positive aggregate balances are changing by the same amount – which is always the case in your 4 examples. Gross aggregate balances can be unchanged (examples 1 and 2), higher (example 3) or lower (example 4).

    Now – back to the “real world” – which is a “mixed world” of loans and negative money balances - the money supply is still the totality of positive balances – which is equal to the totality of negative balances plus (total positive balances less those positive balances equal to negative balances). I.e. money supply can be viewed as either total gross positive balances, or the sum of total gross negative balances plus a defined residual subset of positive balances. That residual subset is the differential of the total and the subset of positive balances that have been created by negative balances.

    (This type of logic reminds me of S = I + (S – I))

    The key point in either the real mixed world or the pure non-loan world is not to double count negative balances with the equivalent amount of positive balances that offset them.

    ... cont'd

  5. cont'd ....

    Regarding velocity – I think in your real world model there is an underpinning issue of stock/flow consistency.

    On the one hand, we have outstanding gross positive and negative balances as balance sheet stocks. On the other, we have a flow of funds that connects to changes in those balances.

    I would say that the changes in balance sheet stocks can be positive or negative with either/both positive or negative balances as starting positions.

    But that change in stocks is the balance sheet result of the flow of funds – which is not precisely the same as the flow of funds itself.

    I’d say from a purely logical/math standpoint that the flow of funds in your real world connected model ALWAYS should be considered as a flow of positive balances.

    All players are transacting in positive money balances in your model – even if they don’t hold them at the start as an asset.

    For example, even in your case 2, which consists of both payer and payee negative balances (start and for illustration purposes – finish), I would say that the flow of funds is defined in terms of positive money balances, as follows:

    Suppose the payment is Z.

    And negative balances are (X) and (Y).

    Then the depositor of (X) pays the depositor of (Y) a positive amount (balance) of Z, and the result is {(X – Z), (Y + Z)}.

    Even if both those resulting balances are still negative, the payment has been in the form of positive balances Z – not negative balances. Just because a depositor starts out with a stock of negative balances doesn’t mean he isn’t transferring positive balances. Positive balances subtracted (paid) from negative balances result in greater negative balances; positive balances added (received) into negative balances may still result in negative balances of a lower amount if the positive payment is less than the absolute value of the starting balance.

    So the velocity is measuring the movement of positive money balances, and velocity itself it can be measured using a money supply that can be measured in terms of the outstanding stock of gross positive balances, or in terms of the sum of outstanding negative balances and the differential between the totality of gross positive balances and that total negative stock. The key is not to double count the negative in the recognition of money supply – because it already exists as positive.

    Regarding your final point, it seems to me that your overdraft limit and your unutilized credit card limit are both loan facilities – not loans. If inside money for example is mostly created by actual loans and destroyed by the repayment of loans, I’m not sure how consistent or useful it is to include the idea of a reserve that has not yet been activated in the measure of money. If that were the case, we wouldn’t bother tracking the actual drawdown or repayments of loans at all in the measure of corresponding money – we’d just add up all the credit limits and ignore actual drawdown and repayment activity and the actual balances that are being created or destroyed dynamically in the process. That doesn’t seem right to me. It seems to negate the whole point of being interested in how much money is outstanding now – not how much there will be in a completely different scenario of money creation.

    That’s your real world connected model in my view.

    I think Nick R’s model is a lot more complicated conceptually, because he’s using outside money only. I may try and see if I can translate that in my own way, if I have time.

    1. Okay. You've got my head spinning now.

      First off, the purpose of the post was to illustrate some of the difficulties with the concept of velocity, specifically as a measure of the circulation of some aggregate. I borrowed the term "negative money" from Nick's post, but along with velocity, it is not a concept that I would normally choose to use myself to explain things in the real world.

      I'm not sure I follow the first bit of your comment about gross positive balances offsetting gross negative balances. I would say though that I think, in theory, an economy could operate with a non-positive money supply. This would involve 100% capital funded banks, which offered a mix of term loans and overdraft. (A few more bits are needed to make this work - but that's the essence.) In this case there would appear to be no gross positive balance (of money type assets), and the conventional measure of the money supply would be zero. I'm not sure if this relates to your point.

      On the latter part of your comment, I would agree that there are good reasons for preferring an analysis that assigns the same direction to the payment, regardless of the sign of the balances to and from which it is made. This is particularly so, if we note that sometimes there is no objective truth as to what sign the balances are when the payment is made.

      In terms of flow of funds, I guess this would create some ambiguity if we were trying to produce these on a highly disaggregated level. So if we made a purchase, we might be unsure whether to allocate the sources element to a reduction of money holdings or to an increase in overdraft. In practice though, this should never matter, as you will always get the same answer once you have aggregated to the level of a single day.

      Regarding velocity again, obviously you can measure the ratio between the total value of payments and the aggregate positive balance measure, and this is what is usually done. My point, though, in this post was to highlight this is no longer an obvious measure of something circulating, because many of those payments do not involve positive balances at all. It would be like taking the ratio between all payments made in the economy and total balances held at just a single bank. It means something, but not really circulation.

      The relevance of this is that Nick Rowe, for example, uses the idea of circulation to distinguish between money and land, say, in looking at their ratios to the level of activity. Of course, I completely recognise that there is a special importance in the relationship between transaction account balances and payment levels. My aim here has just been to point out the danger in reading too much into this, to shed some light perhaps on the reasons why velocity might be an unreliable indicator.

      I accept your point on my analysis of undrawn limits. This was intended just to provoke a bit more thought, certainly not as a workable alternative to current measures. I would note however that I think maybe the role of committed liquidity has not been given enough attention in the analysis of money demand functions. I have heard it suggested (but have not looked into it) that current corporate cash hoarding is connected in part to the reduced availability of liquidity facilities.

      Overall, though all this stuff about velocity and negative money is just a diversion for me. It would not appear in my normal preferred approach to understanding the world.

    2. It’s a diversion for me too.

      I think it may be a form of penance for having described the subject as “nonsense”.

      I’m actually thinking of doing a post on Nick R’s post as a way of completing my process of excruciation.

      The scary thing is that I’m actually finding it interesting now. I’ll have to be careful – or I could have my Monetary Realism licence revoked.



      My point on stock flow consistency was a general one – not that you hadn’t paid attention to it. Signs are important in setting out one’s overall approach, and I have my own preference for doing it. That’s all.

    3. Hahaha.. I've got dibs on the new blog: "Monetary Unrealism" ... it would feature frequent guest posts by Nick. There'd be broad categories like "Cows" and "Apples" and "Haircuts" but of course we could all get in on the act:

      "As we discussed last week in the typical haircut based economy with negative money, blah blah..." Haha. Great fun.

      Or how about "Monetary Complexity" in which money takes on a complex value:

      How much is it then?

      That'll be $(3 + 4i) sir.

      Shoot! I've got $(6 + 2i)... more than enough magnitude, but off in phase again!

      Hey once we start talking about negative money, you had to know that complex money was right around the corner, right?

      It'll sure make interest rates interesting... and sales

      This weekend only! Prices will be divided by (1.4 - 1.4i)! Hurry while it lasts!

    4. Very good.

      Complex money - I like that. Maybe not more outlandish, though, than some other stuff economists come up with.

  6. my post:

  7. Well I agree with you that to get some kind of meaningful measure we would have to aggregate positive and negative dollars as if they were alternative currencies.

  8. This all seems right.

    I think one way of loping at it is that monetary Walrasianism (that's Perry Mehrling's term) wants to analyze finance as if balance sheets only had an asset side. So then Nick R. wants to treat debt as an asset with a negative yield. But they are not equivalent: a unit can issue new liabilities but it can't create new assets on the fly, even negative-yield assets. In a world where payment can be made by issuing new liabilities, "money" would have to include the unused borrowing capacity of all economic units.

    Just discovered your blog, by the way. A lot of good stuff here.

    1. I've not seen Mehrling's stuff on this, so I'm not sure what exactly he has in mind. I would say, however, that I do find Tobin's general equilibrium approach to financial markets ( an extremely useful tool, which I use regularly for thinking about these things. I think of this as being somewhat Walrasian.

      However, it does need more in order to be able to properly represent liquidity and hence "money", which I think relates to the point you're making.

      Thanks for the comment on the blog. You might find one or two bits where my thinking has been influenced by posts of your own.

  9. The problem, I should add, is that it's not just the unutilized balance in your credit card. It's also the unutilized balance on any other credit card you could get approved for, the student loan or auto loan or home equity line of credit you might be eligible for, etc. In a world where payment can be made by issuing new liabilities "money" includes all the possible liabilities you might get someone to accept but haven't yet. No reported number captures that.

    Which means the next step, we have to throw out the whole market metaphor, stop talking about endowments.

    1. I agree with this. In the post, I just wanted to focus on what the available spending power was right there at that time, rather than bringing in questions of how easily one could liquidate other assets or arrange other facilities. However, reading your comment, it occurred to me that these days shops will often offer you a "store card" there and then as they can do their credit checks online, so even what I set out doesn't really cover it.