Friday, August 21, 2015

Information equilibrium as an economic principle


I have finished the first public draft of the information equilibrium paper I started to write back in February. Here is a link (please let me know if it doesn't work -- I think I've set my Google Drive settings properly) and the outline below:

[Now a pre-print on arXiv. Updated the draft paper with edits from Peter Fielitz, Guenter Borchardt and Tom Brown. Version from 9/26/2015 is available here. Previous version was 8/21/2015 and is available here.]

Information equilibrium as an economic principle
http://arxiv.org/abs/1510.02435 [q-fin.EC]
http://econpapers.repec.org/RePEc:arx:papers:1510.02435
http://ssrn.com/abstract=2894072

1 Introduction
2 Information equilibrium
   2.1 Supply and demand
   2.2 Physical analogy
   2.3 Alternative motivation

3 Macroeconomics
   3.1 AD-AS model
   3.2 Labor market and Okun's law
   3.3 IS-LM model and interest rates
   3.4 Price level and inflation
   3.5 Solow-Swan growth model
   3.6 A note on constructing models
   3.7 Summary

4 Statistical economics
   4.1 Entropic forces and emergent properties
5 Summary and conclusion

40 comments:

  1. Looks awesome Jason! Now you should have some time to look over my decidedly less ambitious paper.

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    1. Cheers, Todd. And yes, I am going to have a look hopefully by early next week ...

      Delete
  2. Hi Jason, great!

    First off, in the abstract, should it be:

    "We recover the properties of the traditional..."

    I think you're missing the "of"

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  3. "From this intuition Lee Smolin (2009) suggested a new discipline of statistical economics to study of the collective behavior of economies composed of large numbers of economic agents."

    Seems awkward to me, particularly this bit "to study of"

    I'd either remove the "of" or insert a "the" to make it "to the study of"

    Or perhaps better yet, something like this:

    "From this intuition Lee Smolin (2009) suggested a new discipline of statistical economics for the study of the collective behavior of economies composed of large numbers of economic agents."

    or

    "From this intuition Lee Smolin (2009) suggested a new discipline of statistical economics: the study of the collective behavior of economies composed of large numbers of economic agents."

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    1. or "...statistical economics comprising the study of the collective behavior..."

      and a bit further down, perhaps include another "for" here:

      "...for systems away from equilibrium or for non-physical systems."

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    2. "...in which he applied rigorous approach to economic equilibrium."
      perhaps change to:
      "...in which he applied a rigorous approach to economic equilibrium."

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

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  5. Search for "bee understood" and replace with "be understood"

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  6. Search for "This follows the notation one of the earlier versions" and replace with "This follows the notation of one of the earlier versions"

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  7. Just prior to equation 1, you write "(defining k = " and then you have an expression with "s" (the number of symbols), but you haven't told us the number of symbols for this example with the rod. Or you haven't justified that it doesn't matter for some reason (if it doesn't).

    Also you define q as the length of the rod, but you don't define what u is, nor delta u.

    And we can guess the definition of nu (that is nu = u / (delta u)), but still, it's not made explicit prior to using nu for the 1st time (just prior to equation 2).

    Also on this same page (4), in the equation for p (between eqs. 1 & 2), what does the word "detector" do on the bottom here:

    p = (delta q / delta u) detector

    How would the above expression be different in meaning if the word "detector" wasn't there?

    It's at this point that the level of abstraction has multiplied considerably (page 4). The equations and math is not difficult, but what starts off as a promising simple example (a rod of length q divided into delta q segments), quickly leaves terra firma, and what u, delta u, and p represent in terms of physical things (like rods) becomes hard to fathom. For me anyway. I'd like to see this simple example fleshed out more with a few words about what these concepts could represent in the physical world (like rods, etc).

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    1. Also, is s the same on both the q side and the u side? Does it have to be?

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    2. Thanks for taking the time to read and review. And I agree I should add a few sentences around the initial derivation of the model equation.

      And the s's aren't the same, so I should make them different in the paper.

      Delete
    3. No problem Jason. I plan to return to the task later (because I'm nowhere near done even reading it). I figure if your paper is understandable to me, then it's understandable to anyone with even a modest mathematical or technical background.

      Delete
    4. We had almost this same conversation about that rod example once before:
      http://informationtransfereconomics.blogspot.com/2013/04/the-information-transfer-model.html

      Delete
  8. Jason, in your paper, Figure 2 (b) does not appear to show a shift in the "supply curve to the right." It appears to show a shift of the demand curve from the position shown in Figure 2 (a) (represented by a dotted red line in Figure 2 (b)) to a new position to the right of the dotted red line (the solid red line). The blue curve is labeled with a blue S, but in the text you say the blue curve is the demand curve. The red curve is labeled with a red D. The blue curve doesn't look like it shifted at all.

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  9. Also, in eq. 5, the variable on integration on the RHS is S', but in the unlabeled equation between eq. 11 and eq. 12 (call it eq. 11.5), the variable of integration is S. Why the change?

    Also, in eq. 11.5, why did you insert parenthesis around D' on the LHS and S on the RHS in the expressions for differential D and differential S. In other words, in eq. 5 you wrote: dD' and dS' and this changed to d(D') and d(S) in eq. 11.5. Why?

    Also, on pg. 5 you write "We are now going to search for functions..." and you give two forms, but I don't see where you find precisely those forms in the following development.

    Also, eq. 8 starts with eq. 3, but I don't see where you justify adding the ensemble average brackets around D and S in eq. 8. I think I see what you mean, but this might be confusing for people.

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    1. The parentheses are irrelevant.

      We actually do find two functions of < S > and < D >, but they are subsequently solved to eliminate < S > and < D > as those variables (actually, just the upper limits of integration) parameterize your position along the demand/supply curves.

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  10. In eq. 5 you introduce Dref and Sref, but you don't say anything about them in the text. Are they arbitrary?

    Also I notice that the ensemble average S and D labels in Fig 2 seem to be associated with particular points on each curve. Why?

    I looked at Fig 2 and the text again, and I don't see how there can't be at least one error, since you explicitly say the blue curve shifts, but the blue curve doesn't shift.

    "In the figure we show a shift in the demand curve (blue) to the right"

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    1. Yes, Sref and Dref are arbitrary. This is how you look at gasses in thermodynamics and represent your initial conditions. Sref would be V_{A} in this Wikipedia article on isothermal processes:

      https://en.wikipedia.org/wiki/Isothermal_process

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    2. But we assume that S=Sref when D=Dref for the general equilibrium solution, right? (i.e. the point {Sref,Dref} lies on the GE solution curve).

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    3. It would because

      (D/Dref) = (S/Sref)^k

      so {Dref, Sref} is always a solution since 1 = 1^k for real k.

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  11. OK, I revisited your paper (after having read your April 2013 post about supply and demand). Here's the sentence that's in error:

    "Inthefigureweshowashiftinthedemandcurve(blue)totheright.Thenew equilibrium price is the intersection of the new displaced demand curve and the unchanged supply curve."

    What you actually show is a shift in the SUPPLY curve (RED) in the right hand plot (Fig. 2b), from it's nominal location at S0 to a new curve at S0+delta. (w/ delta > 0).

    The figure labels are accurate.

    Also, it might be confusing for the reader that you labeled a point on the solid red supply curve (a point at x=-0.1), as [D]. This makes sense if you're saying that for a given delta_S we have in the supply curve an expression for [D] (that can be modified to give P on the y axis). But it's not very clear why you labled it like that.

    For the solid blue demand curve you do the same except it appears you label the point as [S]. This makes less sense because although in developing the demand curve you derive an expression for [S] in terms of delta_D, P goes as the reciprocal of [S].

    Also you don't mention in the text why those labels are there or why there are for an x-axis value of -0.1.

    Also, it would probably be nice to mention what you've chosen for Dref, Sref, D0 and S0.

    Also, I don't know what you mean by dP/dt is proportional to S - D (when you mention that you can add an optional time dependence). I saw that in your April '13 blog posts too, but it wasn't explained there either.

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    1. Thanks Tom. Yes, there is some mismatch because I copied and pasted the text from the blog post but changed the figures.

      Regarding the dP/dt -- that is a simple model of price movements that I saw in some economics lecture notes that I thought people well-versed in economics would understand. Kind of like how physicists are generally familiar with the harmonic oscillator ...

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  12. Also, I'm embarrassed to say that until I remember that economists put the independent variable on the y-axis (Price) and the dependent variable on the x-axis (quantity), the verbal logic of supply and command curves never makes sense: not the intersection, but just the curves in isolation:

    "As the quantity demanded increases, the price goes down"... wait, what?

    It makes sense when you turn it around:

    "As the price goes up, the quantity demanded goes down, and as the price goes down, quantity demanded goes up."

    Likewise for the supply curve.

    Now folding in both curves and their intersection, it makes sense to say as "demand increases" (i.e. the whole demand curve goes to the right) for a fixed supply (supply curve that is), then the equilibrium quantity demanded (crossing point) and the price both go up.

    The way you write the equations on pg. 6 and 7 reflects this way of thinking about it.... but for a reader like me who sometimes forgets that economists do this (put the independent variable P on the y-axis) it seems unnatural at first. I'm sure it won't confuse any econ people, but I'm not sure where you're going to submit this thing.

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    1. Yes, I actually had the exact same confusion on my most recent post about Sumner's model that led to the sign error ...

      http://informationtransfereconomics.blogspot.com/2015/08/scott-sumners-information-equilibrium.html

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    2. Also, on page 5 where you say you're going to look for functions F, such that

      [S] = F(D)

      and

      [D] = F(S)

      that's true, and that makes sense because in each case the [X] value relates to P. However, in keeping with economics tradition, you really find the opposite expressions:

      Finv([S]) = D and Finv([D]) = S (i.e. what you write down on pages 6. and 7.)

      This is a minor point, but it doesn't take much to confuse me. Ha! So I don't know if I'd change anything... I'm just giving you my experience in going through it. Perhaps it's not an issue for anyone else.

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  13. Jason, I've got a question regarding section 2.2 (Physical analogy) of the draft paper on pg. 8. You write:

    p*V = (1/2)*N*k_{B}*T

    I'll simplify by writing for k_{B} the following: kB (Boltzmann's constant).

    This Wikipedia page:
    https://en.wikipedia.org/wiki/Ideal_gas

    has:

    P*V = n*R*T

    and it identifies

    U = c*n*R*T = internal energy

    where c is the dimensionless specific head capacity (it's really written with a "^" on top and a "v" subscript, but I'll use "c" for simplicity). And it says that typical values for c are 3/2 (monatomic gas), 5/2 (diatomic gas), and 3 (more complex molecules). It also writes this following:

    n*R = N*kB

    Where N is the number of gas molecules (whereas "n" above is the moles of gas).

    Thus that leads me to conclude that

    P*V = N*kB*T = U/c

    but that doesn't seem to match your equation. Also, matching up the (1/c) with you 1/2 in this expression, I don't see c = 2 as a possibility on that list in Wikipedia.

    Now it's true that Wikipedia uses "P" for pressure while you use "p," but I didn't think that mattered.

    Why don't the equations match? Or did I make a mistake here?

    Also, the Wikipedia page on the Boltzmann constant seems to agree with what I wrote above:
    https://en.wikipedia.org/wiki/Boltzmann_constant

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    1. If the above is correct, that would make the analogy look like this, wouldn't it?:

      Economics:

      price = P = dD/dS = k*D/S (with typical values of k ranging from 1 (Japan) to 2 (QTM)

      physics:

      pressure = P = dU/dV = (1/c)*U/V, with (1/c) ranging from 2/3 (monatomic) to 2/5 (diatomic) to 1/3 (more complex molecules).

      Where did I go wrong?

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    2. It'd be a one dimensional ideal gas in the case of 1/2. For three dimensions you add up three factors of 1/2. The 1/2 comes from integrating a normal distribution. There is (1/2)*kT per degree of freedom, so for a monatomic gas (spherical symmetry) you have 3 spatial translational modes. For a diatomic gas, you add a vibrational mode and a rotational mode.

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    3. In the IT model, we don't necessarily have an equipartition theorem quite yet (I haven't derived one -- it critically depends on the distribution you use for the analog of "energy"), so kappa is a free parameter.

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    4. Jason, will I go wrong if I use the following analogy:

      Economics:

      price = P = dD/dS = k*D/S (with typical values of k ranging from 1 (Japan) to 2 (QTM)

      Physics:

      pressure = P = dU/dV = (1/c)*U/V

      I have a hard time imagining a "one dimensional gas." Is that a gas in which the molecules can only move in 1-dimension? In the case of a cylinder with a piston, say along the axis the piston is free to move? If that's the case, then the piston experiences a pressure of (1/2)*N*kB*T? I kinda get why the constant is (1/2) for a one dimensional gas (if I'm interpreting that correctly), but what I don't get is why in 3-dimensions they leave the "degree of freedom" based parameter out of the ideal gas law (i.e. they write it P*V = N*kB*T = (1/c)*U).

      Also, it sounds to me like for the case of a one dimensional gas, the energy is not N*kB*T but (1/2)*N*kB*T, given that the constant is included in the definition of U in Wikipedia.

      However, in the draft paper you have "the energy N*kB*T is ..."

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    5. "what I don't get is why in 3-dimensions they leave the "degree of freedom" based parameter out of the ideal gas law (i.e. they write it P*V = N*kB*T = (1/c)*U)."

      I should have said I don't get why Wikipedia does that but you don't. I guess the Wikipedia version makes more sense to me right now. But maybe there's no conflict and I'm just missing something.

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    6. As a physicist you quickly learn to discard various factors of 2 ... it seems right on Wikipedia:

      https://en.wikipedia.org/wiki/Equipartition_theorem#Ideal_gas_law

      A one dimensional gas would exist in a one-dimensional world -- it is a mathematical construct. The atoms would somehow have to travel through each other (since there is no way around them) which makes the assumption of zero volume fail.

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    7. Ah, OK. I was going to make a joke about one dimensional gas being a digestive byproduct of spherical cows in a vacuum.

      You might consider adding a word or two to clarify that in the paper, in case somebody like me (i.e. easily confused) decides to dig into it.

      The whole reason I did it was because I wanted to write a "perfect" analogy, and I think using "U" and "dU" works... even the constant (1/c) works out nice, except that it ends up (typically) on (0,1] rather than the typical interval for k (i.e. [1,2]).

      Delete
  14. Jason, I noticed that on page 24, section 3.6, first bullet point you write:

    "In the Solow-Swan model, we used N ⇄ K and N ⇄ K to define the production function."

    Should that be

    "In the Solow-Swan model, we used N ⇄ K and N ⇄ L to define the production function."

    ?

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    1. Also I'm pretty sure the figure numbers for the three bullet points on pg 24, section 3.6 are wrong. The 1st bullet point should refer to figure 13, the second to figure 14, and I don't know about the 3rd (are you missing a figure?).

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  15. Very cool to see you apply Verlinde to sticky prices. I have to read the specifics but seems like a very interesting idea.

    You do not cite Duncan Foley et al. Foley was chair of economics at the New School and has published a fair amount of work using thermodynamics to derive general equilibrium economics. His approach seems largely congruent with yours.

    You can look up "thermodynamics Duncan Foley" in Google Scholar. As a shortcut one physicist friendly PDF is at (http://www.santafe.edu/~desmith/PDF_pubs/DYNCON2011.pdf).

    You might find the folks at the Santa Fe Institute more receptive than most economists, by the way.

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    1. Thanks for the link -- I have read Foley's work (I reference it here for example).

      Originally, I wanted to see how far I could get while avoiding introducing "utility" (which Foley bases his work on) ... but since that time I've become a bit more amenable to utility as an "effective field", so maybe it's time for a revisit.

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