Thursday, January 22, 2015

A Socratic dialog on information equilibrium in economics

info: Hi econ! I'm working on a new economic theory -- can I get some input?

econ: Why are physicists always drawn to economics? You probably just want to show off how good you are at math again. I'll bite. What's your theory, info?

info: Actually, the math is really simple. Simpler than a DSGE model, anyway. It's based on information theory, and ...

econ: Information theory? Economists already use information in their theories. There's a whole field of information economics ... it's even got its own Wikipedia page.

info: That's not exactly the same thing. The content of a piece of information matters in information economics, right?

econ: I don't follow. How does a piece of information differ from its content? Information is its content.

info: There is a difference between saying you have a 10 GB flash drive and saying you have a 10 GB image file of a kitten, right? The first is only a quantity of information and the second is a specific instance of that quantity of information.

econ: I see. But I still don't see how that makes your information theory so different from information in ... oh, let's say game theory.

info: That's a good place to see the difference! In a game with perfect information, like chess, the locations of the pieces matter ...

econ: Of course the locations matter. It wouldn't be chess if they didn't.

info: Well, in the information theory I'm talking about, they don't. Actually, it's more like they might matter, but we don't really care. Anyway ... the information in the chess board can be encoded in a number less than 13^64 (six possible pieces of each color, plus no piece, in 64 squares) that would take about 237 bits of storage. That even includes impossible positions, such as a rook on every square.

econ: How could the impossible positions even matter?

info: That's a good question, but the only reason we know that some positions are impossible is that we know the microfoundations of chess -- the rules of the game. We don't really know the microfoundations of macroeconomics, yet.

econ: So you're not making any assumptions about the microfoundations of chess, you're just looking at it as 237 bits going back and forth between players?

info: Exactly! That's information equilibrium. The information received is the information sent. That's a great analogy ... I'll have to remember that one.

econ: You're welcome.

info: There's also something detecting those 237 bits going back and forth ...

econ: The win-loss ratio?

info: You're good! It would be the win-loss ratio of one of the players, though. We'd probably want to take a logarithm, too.

econ: Naturally.

info: If two players are basically equal, or basically making random moves, you'd expect a win-loss ratio of about 50%. If it changes, one player is probably better than the other.

econ: But what does the 237 bits have to do with this?

info: Well, we'll need to consider if the chess board gets bigger or smaller for economic growth.  And if there is a shade obstructing half the view of the board, only 118 bits are getting through. You'd expect a good player's win-loss ratio to fall under those conditions.

econ: What is that supposed to be, a recession?

info: I am not completely sure at this point. I'm still working on the theory.

econ: I'm just looking for the connection to economics.

info: Ok, ok. You can make the analogy of one player winning more than usual as a positive shift in the demand curve, raising the win-loss ratio. That's the price. It could also be a negative shift in the supply curve.

econ: We economists usually say the curves shift right or left, or up or down ... if we even use supply and demand curves at all outside of undergraduate economics 101.

info: So you'd probably not be impressed that information equilibrium leads to supply and demand curves?

econ: Not really.

info: Or the IS-LM model?

econ: Your ad hoc theory lets you come up with ad hoc models? Pass.

info: I've made some predictions of the future path of inflation.

econ: Economics isn't really about predictions. We're more like doctors than The Weather Channel. When something goes wrong with the economy, we tell you how to fix it.

info: Well, the theory allows you see the different effects of fiscal and monetary policy ...

econ: Well that's interesting. And relevant! Let's hear it.

info: Remember the 237 bits in the chess game? Well let's say NGDP is an encoding of a given economic scenario -- a chess board position --- with dollars being something like bits.

econ: Ok.

info: Now lets posit that NGDP is in information equilibrium with another number encoding the same economic scenario. Let's use the money supply.

econ: Which money supply?

info: It doesn't really matter right now.

econ: I assure you, it does matter.

info: We'll figure that out empirically later. Just hear me out for now.

econ: All right.

info: And the price that is detecting the information moving around, keeping both numbers in equilibrium like the win-loss ratio, is the price level.

econ: PCE or CPI? Core or headline?

info: [Sigh] ...

econ: What?

info: They're barely different from each other!

econ: Isn't there a joke about a physicist and a spherical chicken?

info: Anyway, when you put these pieces together you ...

econ: ... get the quantity theory of money. Yes, yes. But that's like a hundred years old and has pretty much been discredited by empirical evidence ... unless you add inflation expectations terms, I guess.

info: But it's not exactly the quantity theory of money. That's just the high inflation limit. As the economy grows, inflation tends to fall and that eventually leads to something that looks like a liquidity trap.

econ: Why?

info: The reason seems to be that as an economy grows, a typical dollar is more likely to be used in a transaction in a low-growth market.

econ: But why is that?

info: Of all the possible economies you can have with an given NGDP, most have a bunch of low growth markets and a few high growth markets ... and that ratio gets bigger as the economy gets bigger.

econ: You still haven't explained why.

info: If you don't make any assumptions about how the economy works, that is just the most likely configuration. There are more ways an economy with a given NGDP can have a bunch of low growth markets than a bunch of high growth markets. It's like there are more ways you can send out a given amount of energy with a bunch of low energy photons than with a few high energy photons. It's a maximum entropy argument. Information entropy. There is an entropic force preventing high inflation that gets stronger with the size of the economy.

econ: That doesn't explain anything. You need to tell a story ... how do the incentives change for firms to raise their prices less when the economy is bigger?

info: You want me to tell you the Calvo fairy gets tired over time? Or can't get around to all of the firms as fast as it used to when the economy was smaller?

econ: At least that's a start.

info: That would be like adding a Goldilocks force to an atomic model that makes the atom move to where the density is not too high or not too low in order to explain diffusion. It's not only unnecessary ... it's actually wrong. Like Calvo pricing. Firms change their prices all the time. Menu costs are trivial.

econ: The Calvo fairy is just an example of a micoeconomic model assumption to get effects that are observed in empirical macro data into the model.

info: Then why don't you put your models up against empirical data?

econ: The data rejects too many good models!

info: What?!

econ: That's a bit of a joke. But there are lots of successful real-world tests of economic models. Auctions use economic theories to produce better outcomes. Did you hear about that prediction of how many people would ride a new BART line in San Francisco?

info: I have. In fact, I believe it used a random utility model that says the utility someone derives from making a choice has a deterministic component plus a random component.

econ: That's the one.

info: How is the random component different from what I am talking about? In a random utility model, people will not always make the best chess move they can think of that we can sometimes predict, but will also make random chess moves that we can't predict.

econ: It's still based on individuals making choices -- we just allow that there is a component we can't observe.

info: In that case the number of choices and the distribution those unobserved choices are drawn from -- the information theory -- become your only macroscopic constraints. Your deterministic piece is like the cell phone metadata and the random piece is like the content of a text message. That is exactly what information theory was designed for!

econ: Macroscopic constraints?

info: Sorry -- physics jargon. Empirically observed values of aggregate data. Like NGDP.

econ: I'm still not convinced. And it seems like you've come up with a quantity theory of money that can have a liquidity trap. This won't go well. I'll show you why. Let me call up a couple of friends. [dialing ... ] Hello? Hey, Paul, Scott, listen to this.

Paul: What's up?

Scott: Go ahead.

econ: You probably had no idea you'd be on the same telecon today. Anyway, so my friend info here has this new economic theory that's basically the quantity theory of money with a liquidity trap. Thoughts? You first, Paul.

Paul: Mainstream macroeconomics already has pretty good models of the liquidity trap. And the IS-LM model is a simple way to try and explain it without going all intertemporal. We don't really need new models unless they tell us something new.

econ: What's your take Scott?

Scott: A liquidity trap is a result of incompetent monetary policy -- if the theory really is a quantity theory in the spirit of Friedman, then why can't the central bank just create expectations of inflation or NGDP growth? A quantity theory liquidity trap is a bit of an oxymoron. Just create expectations of a permanent increase the quantity of base money!

econ: Ok, thanks, guys!

Paul: No problem.

Scott: Talk to you later.

econ: See, info?

info: You're right. I have my work cut out for me. The people who would like the liquidity trap in my model think their models are working just fine, and the people who would like a quantity theory don't believe in a structural liquidity trap.

econ: In economics, it's really hard to tell who is right because the data is uninformative. We mostly try to come up with a set of assumptions we like the most and see what we can derive from that.

info: Isn't there a joke about an economist assuming a can opener? Wait ... did you say "like the most"?!?

econ: Yes. And we as a profession really like the assumption that at the root of all economics is a complex ocean of human decisions and expectations. What the representative agent thinks determines the course of the economy.

info: Isn't there a contradiction between the complexity of human decisions and a story with a representative agent?

econ: Nope.

info: That doesn't sound very scientific. And the statement that macro data is uninformative is dependent of the assumed complexity of your models. If you think the models have a lot of dimensions, like millions of agents or an infinite number of expected paths of NGDP consistent with current conditions, then the data is uninformative. If you think RGDP is an exponential curve with a constant slope on a log chart, then the macro data is completely informative!

econ: But of course the macroeconomy is complicated! People are complicated and the economy is made of people!

info: An oxygen molecule is a really complicated diatomic system of electrons and quarks confined in baryons held together by meson fields, but an ideal gas is really simple. All the details of quantum mechanics, Yang-Mills theories with mass gaps and SU(3)xSU(2)xU(1) symmetry come down to a single number.

econ: But the economy's really complex!

info: It's about 260 J/kg K ...

econ: I said complex!

info: I guess I'll just keep writing on my blog ...

econ: Good luck with your theory! Remember, think: "complex"!

info: More like intractable.

econ: What was that?

info: Nothing.


Credits:

Some of the dialog is based on actual questions from, quotes by or blog posts by Noah Smith, Chris House, Robert Lucas, Scott "Scott" Sumner, Paul "Paul" Krugman, Karthik Athreya, Nick Rowe, Simon Wren-Lewis and Robin Hanson. But many are my own lowly attempts at snark and humor.

An alternate title was "A dialogue concerning two high-GRE disciplines", but I thought that was a bit much.

11 comments:

  1. Wait, a rook on every square is impossible? I thought a bishop on every square was impossible. You can't have four black-square bishops on the board for example. Wait, I'm a little fuzzy actually, if you get your pawn to the other side, could you get two black bishops (i.e. trade the pawn, presumably on a black square, into a lost bishop?)

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    1. Since each side starts with two rooks and 8 pawns that could potentially be promoted (not sure how they get around each other, but leave that out for now), you can't have more than 20 rooks on the board. That means a fortiori 64 rooks impossible.

      And yes, you can get two black-square bishops by promoting a pawn. However, the microfoundations suggest that you would choose to promote to a queen :)

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    2. I see what you're saying now. That makes sense and answers my question about having no pieces there too (I was imagining something different). So you really are calculating a loose upper bound, and there's nothing special about rooks: just an example. Yes, I thought about the queen thing too, but remind me: you can promote you pawn to anything? Not just a piece you've lost? I'll look it up. Thanks.

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    3. "Underpromotion" to a bishop occurs about 0.2% of the time a promotion happens, the motivation being to avoid an immediate stalemate which might occur if the promotion were to a queen.

      http://en.wikipedia.org/wiki/Promotion_%28chess%29

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    4. I remember once promoting to a knight in a game to put the king in check. Thanks for the link.

      Delete
  2. Jason, I enjoyed your dialog. Your frustration comes through. Tell me, when did you interact with Karthik Athreya? I recall trying to direct him and/or his grad student to your blog a while back (back When he wrote a guest post on Noahpinion), but I didn't think anything came of it.

    Also in your chess analogy, why not just eliminate the "no pieces on the board" as impossible rather than add it in? That's never the case, right? There may be other impossible states, but that seems like an easy one to discount.

    Of course I'm sure you've left a few people off of your list of credits at the end (no doubt in the interests of brevity), but didn't you get a similar response from Stephen Williamson, David Andolfatto, Michael Freimuth, Marcus Nunes, Mark Sadowski and JP Koning? They all seem to contribute to that dialog a bit, no? Did you ever get a response from John Cochrane? I know I posted links to your posts on John's blog on several occasions, but I never saw a response.

    I wonder if you wouldn't have more luck getting a big name physicist interested in the ITM rather than an economist? I noticed that Sean Carroll did a "Google hangout" with a couple of philosophers the other day, since he's a "philosophy friendly" physicist. I'm not sure what good that would do (that is getting interest from a cross-discipline physicist), but perhaps it would do something. If the econ people feel threatened by bigger name barbarians at the gate, perhaps it'll force a more thoughtful response at some point. I liked Carroll's insistence on using models in a debate he did with a guy named William Lane Craig. (I hadn't even heard of Carroll until a week ago, but have since become a fan). Here's his philosophy "hangout":

    https://www.youtube.com/watch?v=OTn_4fCDbkY

    On a totally different topic, I watched a youtube video yesterday with Brian Greene hosting four different people who had different interpretations of quantum physics, one of which was Sean Carroll and another was a guy proposing what he called QBism (quantum Bayesian interpretation I think).

    https://www.youtube.com/watch?v=GdqC2bVLesQ

    Do you have an opinion on this? Do you personally favor any of those four views, or something totally different? Is the QBism related at all to the "Quantum Conspiracy" video:

    https://www.youtube.com/watch?v=dEaecUuEqfc

    Also, have you heard of this book:

    http://www.amazon.com/Fields-Color-theory-escaped-Einstein/dp/0473179768

    I was looking for a layman's intro ... and both the author and Sean Carroll (in another video) touted the benefits of QFT over QM. Is it really all they say, or are they overstating the benefits (they both seem to imply that looking at quantum physics through the lens of QFT resolves a lot of seeming paradoxes). Here's a (short) video from the author:

    https://www.youtube.com/watch?v=ywZ5_YfwihI

    Sorry to pollute your econ blog with such stuff, but I've been interested in this topic recently.

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    1. David Glasner too perhaps? (for you list). David Beckworth? Nick Edmonds. Cullen Roche (not an economist, but still you got a skeptical reaction there I think). Have you ever had any interaction with Robert Waldmann, Steve Randy Waldman, J. W. Mason, James Caton, Roger Farmer, Steve Keen or JKH?

      I recall that JP Koning emailed Michael Woodford once and got a reply. Vincent Cate tried it too, but came up with nothing. (c:

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    2. One of my co-workers here (a mathematician) used to write his own econ blog years ago. He hosted it from a machine at his home. He was shocked one day to see that the number of hits he was getting was going through the roof, and it turned out that Brad DeLong had stumbled across one of his pieces and featured it in one of his posts! Lol. What about Brad? Any interaction with him?

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    3. It's not pollution; and it's not like I have hundreds of commenters asking questions :)

      In my statement, I said they were either direct questions or quotes from various sources. I never talked to Lucas or Athreya, I just quoted them (the part about rejecting too many good models and "liking" the assumptions, respectively).

      Sumner (directly) and Krugman (indirectly) were the source of the "story" comment. Noah Smith posted about the BART ridership model. House wrote a post about physicists and economics. In a brief interaction with Robin Hanson via email, he made something like the statement about information economics. Wren-Lewis had a post about economists being more like doctors recently. Rowe asked me point blank about what information had to do with supply and demand. I think that's it for the exact references ... the others are more vague and probably come from an agglomeration of sources. I didn't mean to diss anyone that wasn't listed :)

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    4. And about QFT, really all it is is a consistent marriage of special relativity and quantum mechanics. Quantum mechanics is non-relativistic QFT, and QFT is relativistic quantum mechanics.

      Knowing QFT does tend to make you not care about the 'interpretations' of quantum mechanics. I'd say they're a bit like interpreting supply and demand as utility or the price mechanism as an information aggregation process. At the end of the day, they don't change the calculations you do, just your intuition about them. One source of the many interpretations is the fact that there are several interpretations of probability.

      My own thinking about it led me to believe that quantum mechanics is all about information passing through Rindler horizons, similar to this:

      http://arxiv.org/abs/1005.2739

      If you think about it, most places where the wierd things in quantum mechanics happens are when your particle crosses over a Rindler horizon (basically going into the elsewhere piece of a space time diagram). I even have been working on a derivation of an entropic quantum mechanics from the fluctuation theorem. But the math behind the Liouville equation is hard:

      http://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)

      Delete
  3. info: Hey econ.

    econ: What's up?

    info: I mis-spoke the other day when we were talking.

    econ: Oh yeah?

    info: Yeah, it really should be the rate of wins by one player that should be the price in the chess model.

    econ: Why is that?

    info: Well, both really work, because they're supposed to be equal to each other ... a 50% win loss ratio is a rate of 0.5 wins per game. It's just that the ratio is the right hand side of the equation and the derivative is the left hand side.

    econ: I still don't really get what you're saying.

    info: I know. It just started to bother me.

    econ: You should let it go. People make mistakes all the time. I don't think that piece was what prevented me from getting your theory.

    info: Cheers. Talk to you later.

    ReplyDelete

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