Wednesday, April 24, 2013

The information transfer model


We will describe the economic laws of supply and demand as the result of an information transfer model. Much of the description of the information transfer model follows [1].

Following Shannon [3] we have a system that transfers information $I_q$ from a source $q$ to a destination $u$ (see figure above). Any process can at best transfer complete information, so we know that $I_u \leq I_q$.

We will follow [1] and use the Hartley definition of information $I= K^s n$ where $K^s=K^0 \log s$ where $s$ is the number of symbols and $K^0$ defines the unit of information (e.g. $1/\log 2$ for bits). If we take a rod of length $q$ (process source) and subdivide it in to segments $\delta q$ (process source signal) then $n_q=q/\delta q$ and we get (defining $\kappa = K_{u}^{s}/K_q^s$ the ideal transfer index)
$$(1) \space \kappa \frac{u}{\delta u} \leq \frac{q}{\delta q}$$
Compared to paper [1], we have dropped the absolute values in order to deal with positive quantities $q$, $u$ (and changed some of the notation, e.g. $\Delta q \rightarrow q$).

Now we define a process signal detector that relates the process source signal $\delta q$ emitted from the process source $q$ to a process destination signal $\delta u$ that is detected at the process destination $u$ and delivers an output value:
$$(2) \space p =\left(\frac{\delta q}{\delta u}\right)_\text{detector}$$
If our source and destination are large compared to our signals ($n_q , n_u \gg 1$) we can take $\delta q \rightarrow dq$, we can re-arrange the information transfer condition:
$$(3) \space p=\frac{dq}{du} \leq \frac{1}{\kappa} \frac{q}{u}$$
Next, we derive supply and demand using this model.

References


[1] Information transfer model of natural processes: from the ideal gas law to the distance dependent redshift P. Fielitz, G. Borchardt http://arxiv.org/abs/0905.0610v2
[2] http://en.wikipedia.org/wiki/Gronwall's_inequality
[3] http://en.wikipedia.org/wiki/Noisy_channel_coding_theorem#Mathematical_statement
[4] http://en.wikipedia.org/wiki/Entropic_force
[5] http://en.wikipedia.org/wiki/Sticky_(economics)

5 comments:

  1. OK, this is about as far as I got... I guess I should read [1]. You don't mention it, but nu must be = u/delta_u. You lost me with the "rod of length q." The rest is clear, except where equation 2 came from (you say it's a definition, so I'll accept it). I will dig into [1].

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    1. For some reason it didn't reply on your comment ... See below.

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  2. Yes n_u is u/delta u ... Imagine u is the length of a ruler and delta u is the size of a tick mark. There will then be n_u tick marks on the ruler.

    And yes, (2) is a definition. In general if you have a change in u vs a change in q you'd have a detector p = f(dq, du). The only sensible things you can make are derivatives out of dq's and du's, though.

    In the ideal gas case you have the fact that pressure = dE/dV and in our economics case the change in demand for a marginal addition of supply is the price. NGDP goes up by the price of a pack if gum when you buy that marginal pack of gum.

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    1. "Imagine u is the length of a ruler and delta u is the size of a tick mark. There will then be n_u tick marks on the ruler."

      I did get that part, but I was having trouble working out the rest of an example with q and delta_q on the other end and a "Transfer System" in between. So the same could be said for q and delta_q and n_q, right? Perhaps a meter stick with cm tick marks on one side (q) and a yard stick with 1/4" tick marks on the other (u)... so then what is the "Transfer System" between them? The process of lining them up or something?

      I used to go surveying out in the desert with my Dad when I was a kid. He had this ancient equipment (the model of his transit was produced between something like 1880 and 1920... no kidding!... we looked it up this past Father's Day). So I can imagine looking through his level sight (a telescope with a leveling system on a tripod, so it's guaranteed to be pointing in the plane perpendicular to gravity) and the graduated pole the other surveyor marches around with (holding it vertical) at various points on the property so they can measure the relative elevation of each point and make a topographic map. The graduated pole could be q. The surveyor looking at the pole through the site and deciding which tick mark the horizontal level line in the level sight lines up could be the "Transfer System" and the notebook he writes the measurement down in could be u (with lots of factors contributing to less that perfect information transfer). What would be "p" in this case? delta_u? delta_u = delta_q if no effort is made to interpolate between tick marks, isn't it? In what sense is delta_q / delta_u a "process signal detector" in this case?

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    2. In your great surveying analogy, in some sense the whole apparatus and measurement process would be the information transfer process. The information source is the actual topography and the destination would be a topographic map and the detectors would be the various "conversion factors" (measurements along sight lines) converting elevation into the numbers used to construct the map.

      The cm vs inches rulers is a good framework for describing a couple of different aspects of the model. First off, the "price" (transfer detector) in that case is constant at 2.54 cm/in. That is to say if I try to digitally communicate that I cut Y inches off of a piece of string, then someone receiving that information will cut X = 2.54 Y centimeters off their piece of string (in a system designed to keep our strings the same length [1]) so that dX/dY = 2.54 dY/dY = 2.54. Also note that k = 1 in this case because X/Y = 2.54 Y/Y = 2.54.

      If we start to think about the information being transferred, we can see non-ideal information transfer. In the ideal case, your bitstream is exactly proportional the information you need to transfer since log2 Y ~ log Y. If I just tell someone Y in the above story, they somehow miraculously know Y is a measurement in inches and convert to their centimeters -- which means I need to send units along with my number (getting even more into the weeds, they'd need to know that Y is e.g. a big-endian 32 bit IEEE floating point number, etc). This means my bitstream is going to be longer than log Y at a minimum needs to transferred (including error correction, units). People also use different ways to denote inches (inch, in, ", etc). All of this means the information in the source is greater than or equal to the information in the destination.

      Changing k is a little harder to convey when we're talking about numbers, but if you think about sending the text of an English sentence "fifty two inches" and converting to e.g. Spanish "ciento treinta y dos centímetros" you have changing message length (including units), but also changing alphabets. Spanish has more characters (e.g. the í, but also every other accented vowel along with ñ for a total of 32 as opposed to the English 26 letter alphabet -- here treating accents as different "symbols").

      You'd have to leave metrology to get to areas with changing conversion factors. In an ideal gas, p = dE/dV so that the conversion factor isn't the same when energy is E1 J and the volume is V1 m^3 relative to when the energy is E2 J and the volume is V2 m^3.

      [1] You could see our monetary system as one theoretically designed to make sure everyone and everything gets a fair valuation -- keeping all our strings the same length.

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