Saturday, July 18, 2015

Maybe Paul Romer would be interested in information equilibrium?


Cesar Hidalgo's Why Information Grows.
Paul Romer has recently read Why Information Grows by Cesar Hidalgo and cites it approvingly as an example of how to think like a physicist. I actually learned of the book from Diane Coyle's review and was both interested and a bit afraid that Hidalgo might have scooped me (so I immediately bought a copy). In some really general ways he does say some similar things (q.v. my post here), but he doesn't go as far as the information transfer framework/information equilibrium.

This blog post is intended to see if Romer is interested in going a bit further down the rabbit hole of information theory ...

A short introduction

The original idea behind information equilibrium is from the physicists Peter Fielitz and Guenter Borchardt; I have applied it to economics -- thinking out loud with this blog (and likely making many mistakes as my training is in physics, not economics).

The basic idea is that if you want to say two quantities A and B are related to each other (interest rates and the money supply; capital and output) at a bare minimum the two quantities must obey a pretty simple condition based on a communication channel from A (information source) to B (information destination):

$$
I(A) \geq I(B)
$$

At best, B can only receive all of the information A sends. This is the basis of the information transfer framework and it has two regimes: non-ideal (greater than sign) and ideal (equal sign). Ideal information transfer is also called information equilibrium and is related to the various meanings of equilibrium in economics. From the above condition, Fielitz and Borchardt motivated the differential equation:

$$
p \equiv \frac{dA}{dB} \leq k \; \frac{A}{B}
$$

that effectively says in equilibrium fluctuations in A have to produce informationally equivalent fluctuations in B. On the left hand side, we define the derivative (the quantity that 'detects' the flow of information in Fielitz and Borchardt's paper) as an abstract price. On this blog, I've written this information transfer relationship with the shorthand $p : A \rightarrow B$ (detector/price : source → destination). Equality represents the case of information equilibrium ('economic equilibrium'), but the equilibrium solution also represents a bound on the non-ideal information transfer case (via Gronwall's inequality).

As a mathematical formalism, information equilibrium allows you to come up with a bunch of standard results in economics fairly simply (for example, this derivation of the Euler equation is pretty cool, and non-ideal information transfer let's you escape Vollrath's trilemma on Euler's theorem that Romer discussed on his blog). The intuition behind it however, that human behavior is so complex that it is indistinguishable from randomness except in situations of large scale coordination (like the panics in stock market crashes), does take a bit of getting used to given the dominant paradigm of human decision-making in traditional economics.

As a teaser, here are a few posts that assemble the Solow growth model as an information equilibrium model:

4 comments:

  1. Did you attempt to contact Paul with a link to this post?

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    Replies
    1. I did via Twitter. He did read/quote from one of my posts in the past, so maybe there's a chance.

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  2. I'm not a twitter user, but I ventured over to see if you tried to contact Paul via twitter. I couldn't really tell. But I saw your political coordinates and took the test myself. I came out 2.8% Left, 33.3% Liberal. I used leaving it in the middle to indicate "I don't know." I'm persuadable on many of those and/or my opinions are too difficult to sum up with a one liner. I imagined somebody asking me each question on a phone survey, and I guess that makes me more skeptical of what somebody means precisely by some of those. I probably only had a strong opinion on a half dozen questions. And still I can't imagine voting Republican like I used to sometimes do. I'd like to see a moderator ask the current Republican field if any of the candidates believe in evolution again: if nobody raises their hand (as I'd expect), I will again have to write the whole bunch off as lunatics and/or liars. They might as well stamp "I'm deluded and proud of it" on their forehead or "I'm pretending to be deluded because I'm a sociopath." Either case, that precludes a vote from me and makes it a bit easier to determine where I should focus my attention.

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    Replies
    1. Being persuadable on the Internet is a rare bird, so keep it up. Lots of people seem to be unmovable objects these days.

      Regarding Republicans, I can't say they've made much sense I've ever been aware. But then I grew up in Texas not too far from Ron Paul's district.

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