Saturday, March 15, 2014

How money transfers information


In this post I showed how the information flows in a simple market for apples, but here I'm going to show what I hinted at in the earlier post: money is a tool to transfer information from the demand to the supply.


Let's start with a simple system of $d$ buyers (green) and two suppliers of gray bundles of something. Each sale adds $+\log_{2} d$ bits of information [1] about the size of the market $d$ and the distribution of demand to a supplier. Or at least it would if the supplier had some knowledge of the size of $d$ in the first place. If there was only one supplier, that supplier could use the total amount of goods sold $s$ as an estimate since the total amount of information would have to be less than or equal to the information coming from the buyers (the source), i.e. (dropping the base of the $\log$'s):

$$
\text{(1) } I_{s} = n_{s} \log s \leq n_{d} \log d = I_{d}
$$

You can't get more information from a message than the message contains! However, there is more than one supplier, so each supplier only sees a fraction of the total supply and a fraction of the total demand. If this were all there is to it, then while each transaction could transfer $\log d$ bits of information, the supplier would have no idea. Enter money; now with each sale a supplier acquires a few inherently worthless tokens:


How does this help? Well, because these tokens are available to everyone (either set up by some government or based on custom), the supplier has some idea of how many are out there (let's call it $m$):


Now each sale is accompanied by $n_{m}$ tokens of money, so that each token transfers $+ \log m$ bits of information from the demand to the supply. This monetary system could potentially work well enough so that we can say the information captured by the supply is equal to the demand, thus equation (1) becomes:

$$
\text{(2) }I_{s} = n_{s} \log s = n_{m} \log m = n_{d} \log d = I_{d}
$$

We call this ideal information transfer when we use the equal sign. If we take $n_{s} = S/dS$ where $dS$ is the smallest/infinitesimal unit [2] of supply and likewise $n_{d} = D/dD$ for demand and assume $D, S \gg 1$ (a very large market), we can write:

$$
\frac{S}{dS} \log s = \frac{D}{dD} \log d
$$

$$
\frac{dD}{dS} = \frac{\log d}{\log s} \frac{D}{S}
$$

$$
\text{(3) } \frac{dD}{dS} = \frac{1}{\kappa} \frac{D}{S}
$$

where we've defined the information transfer index $\kappa \equiv \log s/\log d$. The left hand side of equation (3) can be identified with the price $p$ because it is proportional to $dM/dS$, or the rate of change of money for the smallest increase in quantity supplied.

But wait, there's more! See, money can be exchanged for all kinds of goods and services:


This means that the total demand of all goods and services ($AD$, or aggregate demand) is related to the total amount of money ($M$) so that, again assuming ideal information transfer:

$$
\text{(4) } P = \frac{dAD}{dM} = \frac{1}{\kappa} \frac{AD}{M}
$$

where $P$ is an overall measure of the price of all goods and services ($AD$); it's called the price level. The rate of change of the price level over time is inflation. Now for the totally awesome part: we can solve equation (4) for aggregate demand in terms of the money supply. It's a differential equation that can be solved by integration. We re-arrange equation 4 like this:

$$
\text{(5) }\frac{dAD}{AD} = \frac{1}{\kappa} \frac{dM}{M}
$$

and integrate

$$
\int_{AD_{0}}^{AD} \frac{dAD'}{AD'} = \frac{1}{\kappa} \int_{M_{0}}^{M} \frac{dM'}{M'}
$$

$$
\log \frac{AD}{AD_{0}} = \frac{1}{\kappa} \log  \frac{M}{M_{0}}
$$

$$
\frac{AD}{AD_{0}} = \left( \frac{M}{M_{0}}\right)^{1/\kappa}
$$

Using equation (4) again, we have

$$
\text{(6) } P = \frac{1}{\kappa} \frac{AD_{0}}{M_{0}} \left( \frac{M}{M_{0}}\right)^{1/\kappa -1}
$$

If $\kappa = 1/2$, then $P \sim M$, and the price level rises with the money supply, i.e. the quantity theory of money. Awesome, huh? Except the quantity theory doesn't really work that well ($\kappa \sim 0.6$ works better for the US and other countries, except Japan where $\kappa \sim 1$ is a better model). But we left out a big piece: aggregate demand (e.g. NGDP) is measured in the same units as the money supply. And to top it off the money supply is adjusted by the central bank based on economic conditions! This is the picture of the macroeconomy:


What does it mean? It means that $\kappa = \log m/\log d$, or the amount of information transferred from the demand to the supply relative to the amount of information transferred by money, is changing! If we assume this happens somewhat slowly [3], we can transform equation (6) into:

$$
\text{(7) } P = \alpha \frac{1}{\kappa (MB, NGDP)} \left( \frac{MB}{MB_{0}}\right)^{1/\kappa (MB, NGDP) -1}
$$

$$
\kappa (MB, NGDP) = \frac{\log MB/c_{0}}{\log NGDP/c_{0}}
$$

where we've replaced M with the monetary base, AD with NGDP, grouped a bunch of constants as $\alpha$, and introduced a new constant $c_{0}$ since the units of money are arbitrary. We can fit this model to the price level (it works best, see the lower right graph, when the monetary base is actually just the currency component of the monetary base):


Pretty cool, huh? Using this model (the information transfer model or ITM), we seem to get less inflation from a given increase in the money supply as the money supply and the economy get bigger. In fact, it can even go the other way -- in Japan an increase in the money supply (blue) decreases the price level (brown):


The rest of this blog is devoted to exploring this model and the concept of information transfer in economic, even commenting on current events based on these ideas. Have a look around!

Footnotes

[1] You can think of it as a sale discovering the ID number of a buyer. If the ID number is a member of the set {1, 2, 3, 4, ... , d}, then you need $\log_{2} d$ bits to represent the ID number. Thus, a sale transfers $\log_{2} d$ bits from the buyer (demand) to the supplier (supply).  I will drop the subscript 2 for the binary log in the rest of the post.

[2] We are looking at infinitesimal units to see how supply and demand change with respect to each other. for those unfamiliar with this concept, it forms the basis of calculus. And as a physicist, I am given to frequent abuses of notation.

[3] Technically, integrating equation (5) as we did assumes $\kappa$ is independent of $AD$ and $M$. However, if we assume $\kappa$ doesn't change rapidly with $M$ or $NGDP$, then the integration can proceed as shown, but is only an approximation.

20 comments:

  1. Jason, this is a very interesting blog. I'm just scanning through a few of your posts. I'm definitely not absorbing everything right now... but I hope to at some point here. I think I stumbled upon this before... you posted a link at Nick Rowe's site, true?

    ReplyDelete
    Replies
    1. Thanks and yes, I did ... Did you ever get an answer to your question?

      Delete
  2. I am very interested in your blog and really appreciate the clear diagrams etc that you have used to help explain it. I'm not there yet though in terms of absorbing it all so sorry if I'm just being dumb with my questions.

    One thing that worried me is the relationship between NGDP and currency in circulation. I think it is important to bear in mind that NGDP determines how much currency is in circulation and not the other way around. The Fed just keeps the ATM machines full. If NGDP is high, there is more demand for currency so more currency gets into circulation. If spending slows (such as after the end of the holiday season) the commercial banks offload excess currency back to the Fed.
    http://www.newyorkfed.org/aboutthefed/fedpoint/fed01.html

    To that extent, the QTM part of it perhaps risks being a bit of a circular argument but I'm thinking/hoping there is probably a lot more in your idea than that.

    ReplyDelete
    Replies
    1. True -- the mechanics of how the relationship is maintained at the micro level isn't really determined by the model. In this post, for example, I considered the scenario where the Fed sets interest rates and that endogenously determines the money supply:

      http://informationtransfereconomics.blogspot.com/2014/03/nick-rowes-model-of-money-stock.html

      In the information transfer model, several different markets can be transferring information at the same time in different ways, such that e.g. the IS-LM model and the "quantity theory" can be operating simultaneously. Normally, the Fed operates through the interest rate channel. There is potential for a shock to come in "through the IS-LM model" that is separate from the shock caused by changes in the money supply. In this post, I propose the idea that there could be a "Keynesian" shock (IS-LM model) and a "monetarist" shock (money supply) since the monetarist shock is insufficient on its own to account for the 'Great Recession':

      http://informationtransfereconomics.blogspot.com/2014/02/the-fed-caused-great-recession.html

      Again, I am not saying this is definitive. The blog is theoretical speculation in general; I won't get too upset if certain interpretations make no sense when applied to the real world and have to be changed.

      Delete
  3. Added note: when I refer to each coin transferring log m bits of information, I missed the chance to point out that this means that suppliers/businesses only need to know the order of magnitude of m to get an accurate estimate of the information transferred by a given transaction -- a much less daunting task than knowing the exact value of m.

    ReplyDelete
  4. You're doing some original stuff here. I'd really appreciate a cognitive jolt from you in the form of an everyday/intuitive metaphor that more or less captures 'information equilibrium'. I might have something to throw back...but I'm unclear.

    ReplyDelete
    Replies
    1. Not to worry, it's in your draft paper. I'll have a try with that.

      Delete
    2. I've been trying to come up with a good metaphor myself.

      Information equilibrium is a very relaxed definition of "equal". Take two people ("supply" and "demand") rolling ordinary (6-sided) dice. Equal traditionally means both sides roll the same thing. Demand rolls a 4, supply rolls a 4. Demand rolls a 2, supply rolls a 2. That's utility maximizing economics.

      Economists came up with matching theory where as long as they're close, you can say they're equal ... demand rolls a 4, but supply rolls a 3. Close enough.

      Information equilibrium is the state where as long as they're both rolling dice with the same number of sides (you could have one die with spots and another with colors) you can say they're "equal".

      This makes more sense for economics -- you're usually trading money (numbers on one die) for goods (colors on the other die).

      The other key fact is that market failures are built into the framework. Sometimes the information from a die roll from demand doesn't get to supply.

      I have a couple of other posts here:

      http://informationtransfereconomics.blogspot.com/2015/05/the-economic-allocation-problem.html

      http://informationtransfereconomics.blogspot.com/2015/05/utility-maximization-matching-and.html

      Delete
  5. Do you assume the money multiplier or the MMT view (loans create deposits):
    http://bilbo.economicoutlook.net/blog/?p=1623

    It seems like you accept the fractional reserve theory of banking, according to which banks can collectively create money (via the multiplier) but individual banks can't.

    According to the credit creation theory , the money supply is not under control of the central bank.

    The credit creation theory of banking is different: it holds that each individual bank can create money "out of nothing" by the simple act of extending credit.

    When banks make a loan they create a deposit and a loan. No prior cash needed. Banks have access to reserves when needed via the central bank.

    Firstly a bank creates a loan. So the loan is to person A, and firstly the deposit is to person A as well. At that point the bank is still fine and still fully funded.

    What happens then though is that person A wants to pay person B.

    If person B is at the same bank, then there is NO problem. The deposit is switched to person B and the bank is still fully funded.

    The fun starts when person B is at another bank.

    What has to happen that is that bank 2 has to take over the deposit in bank 1 from person A. That increases the assets of bank 2 which then creates a new deposit for person B.

    That’s how payment works. Somebody has to take the place of the original depositor in the source bank before you can create anything in the target bank.

    Of course at that point bank 2 is taking a risk on bank 1 and will expect to be paid by bank 1 an interest rate to compensate for that risk.

    And it also means that if bank 2 isn’t prepared to take a risk on bank 1, that nobody in bank 1 can pay anybody in bank 2.

    It’s this latter point that caused the creation of central banks – to make sure that the payment system clears. The theory being that all the banks trust the central bank ‘in the last resort’ and therefore the central bank can ensure payments always clear.


    Evidence from Professor Richard Werner:

    http://www.sciencedirect.com/science/article/pii/S1057521914001434

    and

    http://www.sciencedirect.com/science/article/pii/S1057521915001477

    ReplyDelete
    Replies
    1. Hello Random,

      You ask:

      Do you assume the money multiplier or the MMT view (loans create deposits)

      This is not an entirely germane question in the information theory view as it assumes output is in information equilibrium with some monetary aggregate (which one is an empirical question), but causality goes both ways. Additional money opens up economic state space (expands the opportunity set) for occupation by agents, but additional occupied states in the economic state space can end up causing money to be printed.

      Empirically it appears the growth rate of physical currency (literally paper bills) sets the scale for nominal output as well as inflation in the short run.

      Although due more recent results, I'm now under the impression that money has nothing to do with inflation or output and it's all about the size of the labor force:

      http://informationtransfereconomics.blogspot.com/2016/01/is-cpi-information-theoretic-measure-of.html

      Money in that view may be an intermediary as described above, but to first order the effects are invisible at the macro scale.

      As an aside: the latter two links in your comment don't seem to be empirical evidence -- there is no explicit model calculation compared to data or quantification of model error.

      I do admit that I don't really understand MMT.

      Delete
  6. "Now each sale is accompanied by nm tokens of money, so that each token transfers +log⁡m bits of information from the demand to the supply"

    The way the 1st part of this sentence is written, it sounds like each "gray bundle" costs nm tokens of money. But if that were the case, then each sale would transfer nm*log2(m) bits of information.

    But your comment above indicates that it's each COIN used in a sale that transfers log2(m) bits of information. That makes sense.

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    Replies
    1. I'm not sure what the issue is here other than dropping the log base 2 as understood -- which I noted at the beginning of the post.

      Delete
    2. You're right: there's no issue. Somehow my brain missed "token" in the post and substituted "sale." Several times in a row.

      Delete
  7. I guess I could interpret "sale" as a sale of a single token in exchange for some number (or fraction) of gray bundles.

    ReplyDelete
    Replies
    1. What!??

      What does that even clarify?

      Delete
    2. This comment has been removed by the author.

      Delete
    3. You're right. I see my mistake (reading comprehension).

      Delete
  8. Very nice! I thought that I had seen this before, but I hadn't.

    Proofreading point. You never say what NGDP means, and you only sort of say what MB means. You introduce MB and then shortly afterwards mention the monetary base, which the reader can guess is the same thing.

    Also, you jump to information transfer equality without saying how you get there. OC, it is not necessary to do so, but I think that it would help the reader if you said more than just that it could happen. A short paragraph would be enough. I think you need to say that approximate equality is normal, not just possible, that prices (bids and asks) give information to sellers and buyers, and that causality can go either way.

    ReplyDelete
  9. How do you see Information Transfer Economics apply to NASA Science? Could NASA become the equivalent of a central bank managing the money supply between Scientists and end-users of Science products (aka actionable information that has economic value)? I can imagine a fully decentralized network of commodity satellites owned by many organizations, scientists and end-users interacting using smart contracts and crypto-currency... but this is quite a jump.
    Would love to get your opinion though.
    Many thanks,
    Pat.

    ReplyDelete

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