Wednesday, June 4, 2014

Money: the unit of information and the medium of information exchange

We're all familiar with "bits" as units of information and "bits" as the medium of information exchange. We intuitively understand how a 1 GB SD card is a medium for exchanging 1 GB, or 8 billion bits of information. Why, then, is the medium of exchange and unit of account functions of money such a hotly debated topic on the econ blogs? For example, these posts by Sumner and Rowe have over 100 comments each as of last check

First, I don't think I have to convince you that money is a medium for exchanging information. By giving you 5 dollars, I am sending you the information "you need to give me 5 dollars worth of goods and services and we're square" or as Narayana Kocherlakota would say "remember to give me 5 dollars worth of goods and services later" in his paper Money is Memory [2]. The money in your bank account is essentially a record of the work you've done. Actually, one of the oldest ways of storing information in the world was likely a form of economic transaction. Accounting is just a very specific form of recording history.

When money is just the medium of exchange, for most practical purposes it really doesn't matter if someone gives you some MZM, M1, M2, MB or even short term treasury bonds (these days); we seem to understand that intuitively. Where I think this gets confusing is when we think of money as defining the unit of information. This messes with our minds in two ways: 1) when money defines the unit of information, it is dependent on how much medium of exchange is out there and 2) what do we mean by money (M1, MZM, M2, etc)?

Let's attack the first question first. I think it's fairly easy for us to see how money can be a varying unit of measurement. A pound sterling doesn't mean the same thing in 1700s as it does today -- that is we have to go back and covert the weird units British people used in the 1700s called "pounds sterling" into the rather practical units British people use today call "pounds sterling". That is to say, we adjust for inflation. But that adjustment requires knowing a lot about the value of goods and services that were available in the 1700s and today.

The information we're familiar with doesn't work this way. A bit today is a bit in a billion years. We don't have to adjust for infor-flation because there are gazillions of GB out there on the internet. The basic unit of information, the bit = $\ln 2$ "nats", or the information in a flip of a fair coin, is as static as mathematics. We can exchange 8 billion bits and the result is the same the same way we exchange 8 bits [1].

In the transfer of economic information, the total amount of money does define the unit of information. To be precise, both the total amount of money and the size of the economy come together to define the unit of information we refer to colloquially as "value". And actually, it's "nominal value", since we'd say a pound of bacon is worth a dollar (money units) in the 1970s and 5 dollars (money units) today.

With constant units of information, changing units is pretty easy. We have

\log_{2} 2 = \ln 2/\ln 2 = 1 \; \text{bit}

or 1 bit = (ln 2 nat)/(ln 2 nat/bit). To deal with variable units of information we need some tools. The base of the logarithm determines the unit of information and we have the relationships

\log_{b} x = \frac{\log_{c} x}{\log_{c} b} = \frac{\log_{e} x}{\log_{e} b} = \frac{\ln x}{\ln b}

If the base of the logarithm $b = 2$ we're measuring information in bits. If it's $b = 8$, we're measuring in bytes. We'd like to measure information in terms of money. That means, if $M$ is the amount of money out there and $NGDP$ defines the size of the economy, the unit of information defined by money is

\log_{M} NGDP = \frac{\ln NGDP}{\ln M}

I call this the information transfer index  here (or actually its inverse$1/\kappa$), but really it just defines the units of information (i.e. "value") and is connected to the price level (which is how you adjust for inflation). I've referred to the changing value of $\kappa$ as the "unit of account effect"; we could call the basic mechanism in the quantity theory where approximately $P \sim M$ the "medium of exchange effect" analogously.

As for my second question above: which monetary aggregate? Let's leave that up to observational evidence. It seems $M$ is the currency component of the monetary base (physical currency including printed bills and coins).

[1] At least if are well below the Bekenstein bound. When we start transferring information on the order of the number of bits defined by the causal horizon of the exchange, we'll need to take general relativity into account.

One interesting side note: if we measure information in terms of a fraction of the information in the causal horizon defined by the Hubble volume instead of bits, the Bekenstein bound would become:

I \leq -\frac{RM}{2 \log \ell_{p}H_{0}}

where the constants are the Planck length and the (inverse) Hubble time. This makes the information time dependent in these "natural" units since the Hubble length c/H0 is based on the age of the universe.

[2] Actually, a better way to describe Money as Memory is to say that "I have 5 dollars because I did something worth 5 dollars in the past that the community owes me for, and I'm discharging the community's obligation because you are giving me 5 dollars worth of goods or services in return".

Update (6/8/2014): Reworded a couple sentences above, added a few lines for clarity and added the second paragraph to the footnote.

Update (6/14/2014): Added the second footnote.


  1. "information transfer index" ... hmmm, ... how about we call this unit of information the "Smit" in honor of you? Ha! (we can put the "h" back on as long as it's silent... it should rhyme with "bit" don't you think?)

    1. Ha!

      I realize there are a couple of typos in the post (like it should be ln 2 nats and ln 2 nats/bit and I left off the word "exchange" at the end of the first sentence. I kinda wrote it in a rush ... I hope the post was still coherent.

    2. Yes, I think it was coherent.

  2. Jason, I don't know if I've asked you this before, and I apologize if I did, but do you think it would be possible to test the ITM in some kind of a laboratory setting? I've always imagined that a series of online, multi-player games involving the accumulation of wealth as one objective might be appropriate for testing economic theories. The game(s) could be run over several months or years even, and new players could be randomly assigned to different worlds where different circumstances prevailed (e.g. different monetary systems or rules were in play). You'd have to have a means to try to control for saboteurs who are just attempting to disrupt the game, and perhaps real cash prizes would have to be offered to the "winners" to provide better motivation. It's a pretty fuzzy thought, so I apologize for my vagueness, but I was wondering if you'd ever considered such a thing.

    1. Actually, I had thought of the reverse of that ... Using the ITM as the economic logic for a game where you play "Fed". But then I thought it would be really dull, and instead considered it might be good to be the economic component of a game like Civilization.

      The MMOG idea is interesting ... Some social scientists and economists try to do such things -- I remember an experiment using an online music store to see what becomes popular.

    2. MMOG?

      "I thought it would be really dull"

      Have you seen this?

      And did you know that the Fed created and sold a board game in the 1970s? If you zoom in, it's clear those folks are having *tons* of fun. :D

    3. Hey, I just noticed that Noah Smith mentioned this very thing recently:

      "Will huge multiplayer online video games give us a laboratory to study recessions?"


      BTW, that was a rather rude comment that Anonymous left for you (but Noah sure seemed to think it was a hoot).

    4. OMG, the existence of that board (bored?) game is awesome.

      I used "MMOG" for massive multiplayer online game which I made up at the time because I couldn't remember the actual acronym, but it turns out that is the correct one and the actual acronym I couldn't remember was MOOC for massive open online courses.

      And yeah, I've noticed a lot of Noahpinion comments are pretty snarky. I have fairly thick skin and a healthy dose of not taking myself too seriously. There are some pretty funny ones later on down that post (especially between what I think are at least two "Anonymous" and commenter Daniel). I almost died laughing when Daniel wrote: "Are you trying to fail the Turing test ?"

      I think, but am not sure, Noah's blog grew out of the Econ Job Market Rumors forum, which is basically like the comment section on Noah's blog.

  3. Jason, do you have a response to Sumner?

    I think I asked you the same question once regarding changing from the dollar to the penny as the UoA and pointing out that nothing would happen. You agreed, and in fact make much the same point in your post here. But I couldn't find my original comment (and more importantly your response).


      Jason, what say you about this backing issue?

    2. I finally managed to get some time to reply to Sumner. I think you basically did it for me ... thanks! His subsequent response about exchanging $100 for every dollar seems like a far-fetched hypothetical, so I settled for practicality.

      Regarding the backing theory, I am pretty sure that 100% backing is wrong -- mostly because the quantity theory of money is at least partially correct. However, I do think "money" needs some kind of "backing" in order for a monetary system to emerge, at least initially. This can be through legal means, threats of force, valuable assets or, in the case of cowrie shells, simply aesthetic reasons.

      Aside: I recently chaperoned a school trip to an overnight camp near a beach. The kids immediately set up "shops" along the beach and used sand dollars as currency (they're rare, but not too rare, pretty, portable, and, I think for the kids, had the word "dollar" in the name). There was a huge surge of inflation (50 sand dollars for some neat piece of driftwood -- where I think they'd only found 5 or 6 at the time) followed by a massive deflation (ok, 3 sand dollars).

      In the information transfer model, neither the backing theory nor the quantity theory are really correct (the latter is a better approximation) because it's not the quantity of medium of exchange or the value of the medium of exchange, it's the information carrying capacity of the medium of exchange.

      However, there is still some room for "backing" arguments. When a monetary system is established (an maybe after inflationary episodes or in the case of monetary "phase transitions" e.g. after WWII), there really does need to be something to establish the scale of the money and lock in the coefficients of the model. Additionally, "backed" money might do a better job of making the information transfer "ideal" -- no information loss in the information transfer from supply to demand. I can't answer these questions right now.

      It seems like backing might become important when the information-carrying capability of money becomes ineffective or questioned. But overall, in a functioning economy, the quantity theory of money is the better approximation.

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