Friday, January 15, 2016

Velocity of money and interest rates

Since it came up in the top shared posts of 2015, I re-read this post and comments on an MZM quantity theory. John Handley left a comment about how this wasn't all that new ...
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.
He gives a couple of references [1, pdf], [2, pdf]; we'll follow the former and look at the cash-credit [CC] models. The key here to understand is the word "tracks", which isn't quite the same as "is a good model for". From [1]:
[Hodrick, Kocherlakota and D. Lucas] find that if they use the basic cash in advance model, whether it be that of Lucas or Svensson, they cannot generate under any plausible scenario enough variation in the velocity of money. For instance, over the last 100 years the velocity of money has had a standard deviation of around 4.5% but with the basic cash in advance model they cannot generate a standard deviation any greater than 0.09%. However, the cash-credit CIA model can generate more plausible numbers for the variability of velocity, with the numbers ranging between 0.6 % and 5.1%. However, to generate these more plausible numbers the authors have to assume very high levels of risk aversion. As a consequence, while they can explain observed volatility in the velocity of money they are unable to account for several other features of the data, such as the low level of actual interest rates.
This is moment matching (getting the same standard deviation for fluctuations); if we take the basic model in the end of [1] with V being velocity and i being the long term interest rate (see also IS-LM model below), there is indeed a linear relationship between V and i:

V = 1/α + i (1 - α)/α

The parameter α represents the relative weighting of utility between goods that can be purchased with cash versus credit. From the text: "This varies positively with the rate of interest ... and so potentially the model is quite realistic." Yes, potentially. The best fit I was able to achieve is this:

If you look at the fluctuations, they actually do track each other. We'll make it more obvious (and make the model work better) by performing a linear transformation on velocity:

a V + b = 1/α + i (1 - α)/α

Now the best fit looks a bit better (showing the [smoothed] fluctuations this time -- the linear transformation is just a scaling of the fluctuations):

They do track each other. The information equilibrium [IE] model still works much better (matching both levels and rates):

The IE model is:

log i = c log(NGDP/MZM) + k

Based on the information equilibrium relationship

(i ⇄ p) : NGDP ⇄ MZM

i.e. the interest rate i is in information equilibrium with the price of money p which is the detector of signals maintaining information equilibrium between NGDP and MZM.

IS-LM model
As John correctly states, the IS-LM's downward sloping LM curve does sort of imply a linear relationship between velocity and the interest rate. Starting with:

M/P = L(i, Y)
Let's say L(i, Y) is separable so that

M/P = Y L(i)
Which means:

V = 1/L(i)
If we take

V = a i + b

L(i) = 1/(a i + b) = (1/b) - a i/b² + ...

... which is a downward sloping curve for a and b positive (as they are). I do put together an empirically accurate form of the IS-LM model in this post that instead uses investment and the monetary base (as opposed to MZM). One important piece of that model is that it looks at the short term interest rate instead of the long term interest rate, which is more relevant to policy as the Fed sets short term interest rates. The information equilibrium relationships are:

(i ⇄ p) : I ⇄ MB

c : NGDP ⇄ I

Where I is investment, MB is the monetary base, p is the price of money, i is the short term interest rate, and c is an irrelevant price in the investment market. Two additional notes:
  • The IS-LM model is valid when the price level does not change rapidly with MB because it makes the assumption that P does not depend strongly on monetary policy so that NGDP = α Y with α constant. This also means there's no significant difference between real and nominal interest rates.
  • You can add a labor market to this through the Solow model, pull out dynamics for the savings rate and add capital.

Update 16 January 2015

With some tweaking I was able to get a pretty decent fit with the modified CC model:


  1. Jason,

    The empirical validity of CC models is highly dependent on the calibration. The relevant result from the model is that the marginal utility of consumption of cash goods is equal to the marginal utility of consumption of credit goods times one plus the opportunity cost of holding money (roughly equal to the nominal interest rate). In this case, the money demand function depends almost entirely on the utility function. The function may be linear, as models with log preferences suggest, or it may be logarithmic like the ITM version of the money demand function.

    I guess what the authors of the paper are referring to in the quote above is that the utility functions that make CC models plausible fail to square with either micro or macro evidence (i.e. agents are too risk adverse in calibrations where the money demand function is empirical). I'm not sure if this serves to invalidate the model wholly. My intuition is that there is a way to at least preserve the good aspects of the model (e.g., the possibility of an empirically valid money demand function as well as the property that money demand becomes indeterminate at the zero lower bound) whilst making up for its shortcomings (requiring implausibly risk averse agents to be empirically valid) without deviating so far from mainstream economic theory, as you do, or making extremely implausible assumptions, as MIUF models do.

    1. As always, well put. And nice insight.

      I managed to do some tweaking and fit the modified CC model (it's just a linear transform of velocity) a bit better, shown in an update.

  2. So using this model can you then model hyperinflation?

    Can you use this to make a different prediction for Japan? (I think your last one was wrong. :-))

    1. Core inflation has only been about 0.5% in Japan since 2013 and the price level remains consistent with the model prediction. Here's the recent update:

      And here was the last update:

      I always thought hyperinflation referred to inflation rates well north of 10% ... we didn't refer to it as hyperinflation the 1970s.

      Or have the hyperinflation goalposts moved so that a burst of 2% inflation due to a 3% VAT tax increase means it's time to buy a wheelbarrow to haul your extra yen for some dagashi?

    2. Those links are not using any relation between velocity of money and interest rates, are they?

    3. Implicitly.

      log P ~ k log(M0)
      log rL ~ c log(NGDP/M0) = c log(V_M0)
      log rS ~ c log(NGDP/MB) = c log(V_MB)

      where MB = monetary base, M0 = MB less reserves

      c is (the same) constant for both rS = short interest rate and rL = long interest rate

      P = price level

      They (P, rL) are related to each other through this diagram


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