## Wednesday, June 11, 2014

Tom Brown and Mike Freimuth discussed the information transfer model on the latter's blog Free Radical. Mike apparently is a PhD candidate at the same school I went to for my PhD. Small world. Anyway, Mike brings up a couple of points that I thought I'd address here.
It seems like he is just saying that the price level is correlated with the size of the money base.
Actually it gives an explicit functional form for the price level:

$$P(M) = \alpha \frac{1}{\kappa} \left( \frac{M}{M_{0}} \right)^{1/\kappa - 1}$$

Where $\alpha$, $M_{0}$ and $\kappa$ are free parameters that can be used to fit the empirical data. Since constant $\kappa$ doesn't work very well,  I tried assuming that $\kappa$ could vary since it is actually based on the potentially changing information content of 1 unit of NGDP vs 1 unit of currency (it is proportional to the ratio of the Hartley information for the demand "states" and supply "states"). This gives us:

$$P(M, N) = \alpha \frac{\log N/c_{0}}{\log M/c_{0}} \left( \frac{M}{M_{0}} \right)^{\frac{\log N/c_{0}}{\log M/c_{0}} - 1}$$

A different motivation of this equation, entirely from long run neutrality of money, is presented at the link here. The model is the simplest model of economics consistent with long run neutrality (homogeneity of degree zero of supply and demand functions).

The model is also a model of supply and demand in general and is capable of constructing many traditional diagram-based micro and macro economic models.
... it just seems like it boils down to MV=PY to me.
It is, but with a specific model for $V$ and $Y$, namely $V = \kappa P$ and $PY \sim M^{1/\kappa}$.
Putting aside the fact that I’m not exactly sure what he means by the information-carrying capability, one must wonder why this wouldn’t be constantly in question. If the theory is that it works because nobody questions it then I find that theory unconvincing.
The theory doesn't explain why money has value (information carrying capacity, i.e. the capability to be used to mediate information transfer). It is agnostic on that. What it explains are the dynamics of money in terms of the size of the economy and the quantity of money. Given a value of money at one time, it gives the value at another (via the price level). Actually, the model can give the price level from 1980-2014 given the data before 1980 -- and you can extrapolate our current low inflation environment starting in the 1970s. See here:

http://informationtransfereconomics.blogspot.com/2014/05/out-of-sample-predictions-with.html

An analogy: Thermodynamics doesn't explain what atoms are (it was actually worked out without knowing what atoms are) -- it explains the dynamics of huge ensembles of atoms. It doesn't even work out e.g. what the volume is for an ideal gas at a given pressure, but rather figures out the relationship between pressure and volume (e.g. $P_{1} V_{1} = P_{2} V_{2}$). Something like backing theory or some other consideration gives money its value just like quantum mechanics gives the theory behind atoms.

Regarding the "questioning", this was probably unwarranted theorizing on my part. There was something that changed in the US and UK (maybe more countries but I don't have data) that corresponded to the end if WWII that made the coefficients in the equation above change. I think it was Bretton-Woods, but the theory also allows for other major changes in currency to reset the coefficients of the model. In terms of the thermodynamics, we have a law like $P V^{\gamma} = \text{ constant}$; certain things can happen (like a chemical reaction) that causes the "constant" to change.

1. Jason, can you expand on what you mean by this a bit:

"homogeneity of degree zero of supply and demand functions"

Thanks

1. I describe it at the beginning of this link

http://informationtransfereconomics.blogspot.com/2014/02/i-quantity-theory-and-effective-field.html

This is a good overview:

http://www.economics.utoronto.ca/osborne/MathTutorial/HOMF.HTM

But it basically says that if you scale up the variables, the result is the same as if you hadn't. This is the mathematical language for "long run neutrality of money".

2. Hey Jason,

Small world indeed. I just wanted to say that when I said I don't know what you mean, it wasn't meant as a criticism of your model, I actually meant that I didn't understand due to a lack of careful investigation on my part. As far as I know it makes perfect sense, I was just commenting on what I could glean from two posts that Tom directed me to, one of which was an empirical test that (if I understood it right) was basically price on money supply.

So I think we are not disagreeing, I was sort of saying the same thing you are which is that what I was trying to explain is why money is valuable in the first place and that isn't what you are trying to explain (or at least I didn't see an explanation of it there). I actually like the way you put it in this post (modeling V as a function of P), although I think there should be some other stuff in there (like interest rates for instance). I have mucked around with something kind of like that a little so I will have to look at what you are doing a bit closer. After I get caught up with the communists--I mean monetary circuit crowd.

3. Hi Mike, I didn't take it as critical -- I just like to have answers if someone has questions.

I do have some interest rate stuff in various places on this blog; this is a good start:

http://informationtransfereconomics.blogspot.com/2014/03/the-effects-that-move-interest-rates.html

2. Another post by Free Radical (Mike Freimuth) entitled "Endogenous or Exogenous Money" this time:

Jason, you and I have discussed these terms ("endogenous" and "exogenous") before and I think I stated that it sounded to me like you used them in a different way than, for example, someone like Sumner might. When Sumner wrote a follow up post to Nick Rowe's post The sense in which the stock of money is "supply-determined" (weirdly I have that title memorized, since I spent so much time on that one) in which he states:

Nick Rowe has a new post that argues the money supply is fully exogenous, even when the central bank is targeting interest rates..

I don't agree that this is what Nick is saying exactly, but Nick didn't have a problem with it, including the word "fully" even when he repeatedly wrote, both in his post, and in comments to me:

"The supply (function) of money, and the demand for loans, together determine the quantity of money created, and that quantity created (eventually) determines the quantity of money demanded."

So, as Nick said several times, it's "both things" .... "together." Mark Sadowski and Frances Coppola eventually clued me in that Scott takes a really wide macro view, and thus anything setting even the envelope of where another variable might bounce around between, he calls "exogenous." Here's how Frances put it:

“Typically, Sumner simply avoids the problem by ignoring the endogeneity and focusing on the exogenous “envelope” which constrains endogenous monetary base creation and interest rates. He’s very “macro”….sometimes I wonder if he realizes that woods are made of trees.” As you can see if you follow the link, this amused Sadowski.

(BTW, I even put up a "trash" post on Nick's article to help myself understand it better)

Wow, that was long tangent... the point was I'm not sure you are using "exogenous" and "endogenous" in precisely the way any of those folks are (Rowe, Coppola, Sumner or Sadowski), least of all Sumner. But I'm not sure, so I'll finally get to my question, after one more brief aside: When you write something like this:

x ~ y

Are you saying x is proportional to y or x is approximately equal to y? If the former, is that always the case? How would you write x is approximately equal to y? (sometimes I think you use it for both meanings, depending on context, but maybe I'm wrong).

OK, so you write something like this:

log P = (1/k - 1) * log M0

Where

k = log M0 / log NGDP

if k = 1, then dP/dM0 = 0, but you say that one way out of this is for the CB to "exogenously" increase M0, right? Perhaps you said to accelerate M0 w/o regard to what the economy is doing, and you called that "exogneously" increasing M0. BTW, was it strictly "M0" here, or would MB work for that? Anyway, this leads to (hopefully a brief bout of) hyperinflation, which gets the economy out of the information/liquidity trap.

But where does the distinction between exogenous increases in M (I'll just write "M" in case it more broadly applies to other monetary aggregate) vs endogenous increases in M come into play in your formulas? I don't think the two formulas I give above are sufficient to capture that mechanism, because it's not clear what NGDP does (if NGDP changes in the right way k could continue to be at 1, and thus dP/dM would continue to be at 0, right?).

1. Shoot, the 1st formula above should have been:

log P ~ (1/k - 1) * log M0

2. Hi Tom -- I'm on travel again so I'm replying via cell phone so excuse the terse language.

First, when I write x ~ y I mean it in the way physicists do :) ... I.e. x is of order y which means x is proportional to y but there may be subleading terms like x = c1 y + c2/y + c3/y^2 + ... In physics it's usually pronounced "goes as".

I have a post on exogenous and endogenous that gives the exact definition I use and I understand the differences. I think the typical usage seems pretty vague ... Maybe it's actually a more precise usage I don't understand the nuances of :)

I think Coppola's concept would be an exogenous trend with endogenous fluctuations. But I'm not sure.

Endo and exo come into the formulas via the solution of the differential equation that gives the formulas.

dN/dM = k N/M has solutions

N ~ M^k if both N and M are endo

N ~ exp(k M/M0) if M is exo (M = M0 set externally ... You can see changes with respect to the externally set value by varying M)

If both are exo, you get the differential equation with the derivative set at the price

p = k N0/M0

Like the interest rate market.

3. Exogenously increasing M means you are changing the formula from N ~ M^k to N ~ exp M. You've set different boundary conditions on the Diff Eq ... You're changing how the system works.

In physics there are electro static solutions to the field equations (Coulomb's law) and there are electrodynamic solutions (radiating waves)

Related concept here.

4. Jason, thanks! That's very helpful.

So the "subleading terrms" are c2/y and c3/(y^2), correct?

I'm going to have to review my electrostatics and electrodynamics. I do have an inkling of what you mean though. Would it be fair to say that the endogenous solution is analogous to the electrostatic equations, and the exogenous one is analogous to the electrodynamic?

How can you tell which set applies in any given situation though? Say a CB, trying to free an economy from a liquidity trap, announces it will buy $X billion a month, and will continue to do so until inflation or NGDP hits a target, and then they will continue with the purchases another N months after the target is reached? Would it be different if they said they'd increase the monthly purchases by$Y billion each month until the target was reached? (but still keep the part about continuing for another N months)? Of course they follow through with their plan exactly. Does their announcing what their intentions are ahead of time make any difference? I'll guess "no" according to the ITM.

I asked David Glasner once (about a year and a half ago) what he'd do if he were running the Fed. To my surprise he answered the question with a pretty specific yet simple plan akin to the latter of the two I sketched out above. I won't get the figures quite right, but he said something like "Announce the CB will buy $50T a month until NGDP is 5% (annual growth), increasing$50T each month, and then continue purchasing at that level for another six months." So it might be a progression like $50T,$100T, $150T, etc, until NGDP (or NGDP futures... it's a bit fuzzy now) hit the target, then stopping the monthly increase, but continuing with the monthly purchases (say at$200T/month) for another six months. Like I say, he was specific, but don't take my numbers too seriously since I don't recall that well. I'm sure he qualified his plan with a "for example" or equivalent.

Now what about what the Fed actually did? Did we get a combination of both kinds of differential equation solutions (exo and endo)? What about what the BoJ did? Perhaps the BoJ was a bit more skewed towards exo? (a guess). Is it possible to tell how much of each kind of solution solely by looking at the data? Again, I'll guess "yes."

Also, wrt electostatic vs electrodynamic, the latter is an explicit function of time, correct? While the former is not? So does the same apply here?

Thanks again! (I can wait till you get back if it's a pain to respond).

5. Jason, my post above looks like it got corrupted somehow a little. I must have inadvertently typed some kind of escape sequence. Hopefully it's still understandable.

6. The part that got messed up was my description of David Glasner's off-the-cuff plan for the CB. And I was able to find his response. Not surprisingly my memory was playing trick on me a bit. Here's what he actually said (not as specific as I imagined):

http://uneasymoney.com/2013/05/14/wherein-i-try-to-help-robert-waldmann-calm-down/#comment-19008

"Tom, Announce that the size of asset purchases will continue to increase until the inflation target it met and will be kept at least that level for a fixed period of time. I have no particular opinion about the asset mix. On fiscal policy, see my reply to Becky above."

7. It's because you typed some dollar signs, which LaTeX's up the result. However, the result came through fine in the notifications.

For your first question, yes, those are the sub-leading terms. The concept is derived from "big O" notation, so that instead of saying f(x) = O(x) which is already an abuse of notation as it should be f(x) ∈ O(x), physicists have a tendency to write f(x) ~ x.

There's more here:

http://informationtransfereconomics.blogspot.com/2013/09/exit-through-hyperinflation.html

In this example, it's not clear when the hyperinflation in Argentina started:
http://informationtransfereconomics.blogspot.com/2014/01/rich-countries-poor-countries-japan-and.html

Regarding the electo dynamic and electro static: yeah, I'd say dynamic is the endogenous N and M (like radiating EM fields, they kind of evolve to produce each other) and M exogenous is like static (like a Coulomb field produced by a static source).

Regarding e.g. Glasner's Fed policy, I'm sure extreme amounts like 50 trillion would break some other assumptions in the model. If it is a much smaller amount, then that is definitely endo M and endo N and in general when the central bank announces an M policy that targets something like N or the price level, you get deflation at large values of M.

Hyperinflation seems to occur when monetary policy is conducted to accomplish some other goal besides macro policy -- like paying for a war/fiscal policy.

It is hard to say in real time which equations apply. There haven't been enough "transitions" in the data (another way -- there haven't been enough different monetary policy regimes in the world) to really narrow down how they transition from one to another. But the good thing is that it is a linear ODE so the solutions form a vector space which means a set of nearby points can be locally approximated by straight lines.

8. Again on the electrostatic vs electrodynamic: it seems that economics has it's own vocabulary for the two classes of analogous solutions (in mainstream econ). Don't they use terms like "equilibrium" and "partial equilibrium?" I'm imagining those two fall under your endo cases. What about for exo? Do they say "disequilibrium process?"

9. Shoot, based on your last comment, it looks like I got exo and endo backwards in the electro analogy. OK, interesting: not what I would have expected. That's a great analogy though... very helpful.

10. Looks like my reply got garbled too.

By "more here", I meant to link to
http://en.wikipedia.org/wiki/Big_o_notation

The next paragraph was about how it is uncertain which solutions apply and gives two examples: the US around WWII and Argentina.

I'm not sure about partial equilibrium vs equilibrium -- I will have to educate myself a bit and think about it. My first take off the top of my head is that when something is exo it's a partial equilibrium. Supply and demand curves take each to be exo in turn, so that is a partial equilibrium, whereas taking both endo is "general" equilibrium.

11. What about "disequilibrium process?" ... to tell you the truth, I can only say for sure that I've seen Bill Woolsey use that one, but I've seen him use it repeatedly, usually in comments on other people's blogs. I'm not a regular reader of Bill, so probably any of these will do (I just Googled his name and that phrase):

Ah, I do recall a specific JP Koning article that he used it, let me look that up:

http://jpkoning.blogspot.com/2014/04/rowe-v-glasner-round-33.html?showComment=1396701015356#c8558474065390393778

(that happens to be a particular JP Koning post I went through in great detail)

http://www.princeton.edu/~erp/ERParchives/archivepdfs/M269.pdf

a "disequilibrium process" is one where the quantity is the min(Qd, Qs) ... it might relate to what I call non-ideal information transfer where information received at the destination (supply) is less that the information transferred from the supply. I'll have to read more about it. I'd never heard of it and first just thought it meant non-equilibrium process.

13. Thanks Jason. O/T: what do you make of this?

http://pragcap.com/why-standard-macro-models-fail-in-times-of-crisis

14. It is probably correct that sudden shifts in expectations (changes in the distribution) will cause a model based on those distributions to fail at times when the distribution changes. If the economy really becomes a different economy frequently enough, you can't really be sure about what's going to happen.

I think it can be summarized as martingales fail when things change.

Their line through the unemployment data is a bit "drawn by hand" -- it follows some peaks and not others ... I drew this line through the UK unemployment data:

http://informationtransfereconomics.blogspot.com/2013/11/the-labour-supply-part-2.html