Friday, September 25, 2015

An MZM quantity theory?

Vincent Cate points out (on his blog) that the velocity of MZM (money with zero maturity) matches up quite well with the 10-year Treasury interest rate (from FRED):

I had actually noticed this before based on a question from Tom Brown, but I hadn't seen the significance regarding the velocity of money in the previous post until Vincent pointed it out. This version of the quantity theory looks like

PY/M = V ~ a i + b

where i is the long term (10-year) nominal interest rate. So the quantity theory model where velocity isn't constant (V ~ c), but rather determined by the interest rate (V ~ i) does look like a successful model that avoids the issues of circularity involving unobservables in my previous post.

Interestingly, this is also an information transfer model where PY = NGDP, M = MZM with detector i, i.e. (i ⇄ p) : NGDP ⇄ MZM such that

c log NGDP/MZM - k = log i
with c = 0.55 and k = 4.27 (see here). 

One could see three different measures of money supply corresponding to three different things

MZM :: long interest rate
M0 :: inflation
MB :: short interest rate

These are also my three favorite money supply measures because they are the least arbitrary. Measures like M1 and M2 include some things (like bank deposits) but not others (like money market funds) because they weren't deemed important at the time. MZM has a rule to determine what goes in (zero maturity) and M0 is physical currency that has a physical reality.

However I do like the simplicity of the single equation for long and short rates in the model I present in the draft paper (as well as the NGDP-M0 path), but really it's up to empirical analysis to determine which is better. (And for what purpose ... policy? forecasts?)


  1. V following i has been part of economic theory for a long time. Heck, it's implicitly in IS-LM (something like log(M/P) = L(y,i) = a log(y) - bi). In more modern macro, there are models ranging from Cash-Credit models (my favorite) to Money in the Utility Function models which most people recognize how terrible this assumption is; but it's mathematically tractable. Every (I hesitate, but I'm pretty sure...) understands that V is not constant and that it empirically tracks interest rates.

    Here's a brief overview of Cash-Credit models:
    Here's a presentation on MIUF models:

    P.S. there are obviously other money demand models, but these are the first that came to mind

    1. Hi John,

      I was unaware of any models where V ~ i; I had thought most economists had shifted over the V is a definition interpretation if they'd even referred to the equation of exchange at all.

      You learn something new every day. Thanks for the links!

    2. Interesting. If people understand that when you lower the interest rate you are lowering the velocity of money, they why are so many people puzzled that there is no inflation? Krugman says "the regular laws of economics don't apply when you are up against the zero lower bound" when he could just say the velocity is really low because the interest rate is really low. I would agree that people realize that the velocity is not always constant but I am not fully convinced that most economists understand it moves with the interest rate. Though clearly some do. I linked to John Hussman as that is where I heard it.

      I see it in the first link, but in the second link I can not find either "interest" or "velocity".

    3. Oh, and Jason very cool by the way!

    4. Vincent,

      Krugman is probably referring to the fact that, in CIA models, the cash-in-advance constraint doesn't bind at the zero lower bound, so monetary expansion doesn't do anything. Alternatively, he may be talking about monetary offset, but that's a different subject all together. If you want to read more on this, read the first couple sections of "It's Baaack" (link:


      The difficulty with the last 10 or so years is that most models abstract from money demand because it's not necessary for price level determinacy since all you really need is a Fisher relation and a Taylor Rule and possibly the occasional government budget constraint.

    5. Vincent cont'd.,

      The quentity theory is just one way of thinking about money demand. Velocity is not an endogenous variable in either of the models I linked, but it can be derived from the FOCs (which is what the author of the first link did). More Keynesian types generally think of money demand (in logs) as m - p = ay - bi whereas monetarists think of money demand as m + v(i) = p + y.

    6. If Japan is spending about twice what they get in taxes, I am guessing they are violating any government budget constraint needed for price stability?

    7. Couldn't Japan use some price instability?

    8. The Fiscal Theory of the Price Level in a nutshell:

      A government can pay for debt two ways: inflation (technically seigniorage) and taxes. If the central bank can't or won't increase inflation, then the fiscal authority can simply refuse to raise the appropriate amount of taxes so that inflation must increase to satisfy the government's budget constraint.

      In the case of Japan, this means that the fiscal authority needs to credibly promise to be irresponsible for there to be inflation. It would help if Japan left its interest rate peg in favor of any other monetary policy, at least until inflation increased.

    9. I believe hyperinflation is a positive feedback loop. If I am right, Japan is headed for hyperinflation not a steady 2% inflation. It is like a pile of gunpowder and trying to make some sparks to keep warm.

    10. Vincent,

      A possible reason for why Japan is not experiencing a hyperinflation right now is that agents still expect that the Japanese government will satisfy the budget constraint some time in the future (it doesn't matter when). To see how this works, take the standard consolidated government budget constraint (b is goverment debt, m is the money supply. τ is the government surplus, ß is the discount factor, π is the gross rate of inflation. All variables are real):

      1. b_t + m_t + τ_t = b_t-1/ß + m_t-1/π_t

      solve for b_t-1:

      2. b_t-1 = ß(b_t + τ_t m_t - m_t-1/π_t)

      and iterate forward to get

      3. b_t-1 = E_t Σß^(1+j)(τ_t+j + m_t+j - m_t-1+j/π_t+j)

      notice that b_t-1 is predetermined; it can't change value this period. This means that the right hand side of the equation must move to match the left. In this case, either future taxes or future seigniorage must adjust the pay down the debt. Assuming the real money supply doesn't explode, seigniorage is roughly equal to the inflation rate, so if surpluses aren't expected at some time to satisfy equation 3, agents will expect a hyperinflation.

      Consider the last twenty years in Japan:

      The debt to GDP ratio increased rapidly, but inflation remained low over this period. This means that the Japanese public clearly expected government surpluses to adjust at some time in the future and pay down the debt. There is no indication that this has changed; so there probably won't be a Japanese hyperinflation any time soon.

    11. "This means that the Japanese public clearly expected government surpluses to adjust at some time in the future and pay down the debt. There is no indication that this has changed; so there probably won't be a Japanese hyperinflation any time soon."

      Sometimes public expectations change really suddenly. If you agree we just need a change in belief and they will have hyperinflation, don't think you are on solid ground to say not "any time soon". A crisis is an "unstable and dangerous situation", so a sudden change is kind of a given really.

  2. A little confused, even as just a definition if the left hand side is a function of i then so is the right hand side....I guess mms might want to de-emphasize looking at interest rates, which is why you got that impression?

    1. I had never looked too deeply into the equation of exchange. I actually had the impression that it was considered archaic in modern economics ... like studying "old quantum theory" in physics.

      I always had the view of interest rates as a price of money, but I just realized that velocity is also a price in the information transfer model:

      price p ≡ dN/dM = k N/M

      N ≡ PY

      "V" ≡ p/k = PY/M

    2. To me it seems like most people thought velocity was just a value that made the equation of exchange true and because of this did not see the use in the formula. I don't think anyone important in economics thinks the formula is wrong.

    3. I guess I got started with a basic money demand function in undergrad, and you are probably right that after the Lucas critique (to the chagrin of even Lucas) studying money demand functions became perceived as somewhat futile. But more importantly all monetary models bake in an equation of exchange. The cash in advance model that we usually derive the Friedman optimal policy rule is an example...the velocity there is held is difficult to model otherwise. Continuous time models allow for much more exciting velocity though! And once you solve for the optima you have predictions about the relationship between money and fundamentals.

    4. But the velocity is clearly not constant once you get the central bank trying wild experiments beyond anything anyone has done in the past, or just by looking at the graph at the top.

  3. Tying the MB and MZM together, there began to be far too many instantly redeemable claims in the banking system for reserves to cover, starting in about 1995. A banking crisis was clearly in the works, and a flood of reserves recapitalized the MZM liabilities in 2008.

    There used to be a very tight connection between reserves and MZM (short and long rates). Reserves-NGDP falling into the rate increase of 2004, alongside steady long rates remaining steady during the interest rate hike cycle, implies a rising MZM/reserve ratio. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain. ;)

    1. Yeah, that basically matches up with the ratio of short rates to long rates:

      The monetary base used to be (currency) + (small constant reserves), but changed over to (currency) + (large reserves).

  4. Jason, I happened to read a comment on David Glasner's blog that had a link to a post by the commentator (philipji) showing the same plot of of velocity and interest rates. He has his own ideas of course (I guess he wrote a book which touches on the subject). I think Philip has commented on your blog before.

    In the comment he says this (as one of his self described controversial ideas):

    "The velocity of money has nothing to do with the speed at which money moves from hand to hand. It depends entirely on the movement of dollars between M1 (currency and M1 deposits) and non-M1 M2 deposits (time deposits etc). Put another way, the dimensions of velocity are not 1/time. Velocity is a pure number and is the ratio of two stocks, not the ratio of a flow to a stock."

    As I recall, you had a post touching on the subject of mixing stocks with flows, didn't you? Can you remember something like that?

    1. It was here:

      However, in the equation of exchange

      P Y = M V

      we have price level P (dimensionless), output Y (real dollars/year), money stock V (dollars) and velocity V. Therefore V must have units of ~ 1/time.

      Philip must have some different equation defining velocity ... but that means velocity is different from economists' velocity.

      Even the Cambridge k version k P Y = M must have k with dimensions ~ time (fraction of output held as cash for the period of the output).

    2. Thanks. Of course you meant to write "money stock M."


Comments are welcome. Please see the Moderation and comment policy.

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.

Note: Only a member of this blog may post a comment.