Friday, February 6, 2015

A simple example of information equilibrium

Gas pump. From wikimedia commons.

In the information transfer model/information equilibrium approach to economics we start with two quantities A(t) and B(t) and ask whether they are in information equilibrium (meaning information is flowing from A to B or B to A).

What is a simple example of two quantities in information equilibrium? The display on a gas pump like the one pictured above. In particular the "total sale" and the "gallons" (or liters) are two quantities in information equilibrium.

You can see the direct relationship with the underlying information theory of communication by squeezing the nozzle in a pattern, say Morse code. The signal you send in gallons pumped is faithfully (i.e. ideally) transferred to the total sale and someone could read that off.

This is actually not just a simple information transfer process, but rather a trivial information transfer process. The price is constant and the variables are linearly related. There isn't much going on here, but it's a great starting point to explore the information transfer model.

So let's imagine your squeezing of the pump nozzle isn't stable and that gas can sometimes be pulled back out of your tank Let's also say the amount of the total sale has an error. The result will look something like this video:

For every tiny (infinitesimal, even) amount the gallons go up, the total sale goes up a tiny amount. The ratio of those two tiny amounts gives you the price at that moment (i.e. the exchange rate of gallons of gas for dollars):

Since the total sale number has an error in it, the price fluctuates a bit (yes, it's a bit expensive for gas in the US). Now let's do a couple of thought experiments.

If you could fix the total sale at some point during your fill-up, the price would go down as the gallons went up. This traces out a demand curve.

If you could fix the gallons at some point during your fill-up, the price would up as the total sale went up. This traces out a supply curve.

So you see how the logic of supply and demand follows from information equilibrium. You can capture it in the formula

(total sale) = (price per gallon) x (gallons sold)

(demand) = (price) x (supply)

Now some economists out there and even some non-economists are probably writing up angry or dismissive comments -- mostly centering on the fact that this completely ignores any aspects of human behavior. It must be wrong!

There are a couple objections we can dismiss right now [1]:
The supply curve is limited to being a line and the price elasticity of supply is always equal to 1! The above used a simple example where the price is constant and the "information transfer index" k is equal to 1. We actually have:

(D) = k (dD/dS) S

Which allows you to get pretty much any shape of supply and demand curves you'd like.

What about units?! We typically measure demand (especially aggregate demand) in dollar amounts, so I don't see the issue here. Supply and demand diagrams are plotted so that the x-axis is quantity supplied/demanded, but at the equilibrium price, they are just proportional to each other -- and scales are rarely shown on a supply and demand diagram.

Supply and demand curves are a lot more complicated than this! Maybe they are. But I can't seem to find any experiments that show what is wrong with the framework described above. See more here [link]. The fact that D = p S is all you need to get supply and demand troubled me too (at the link).

What about shifts in the supply or demand curves?! Take your constant variable D0 → D0 + D1 (for demand curve movement) or S0 → S0 + S1 (for supply curve movement) keeping the other variable constant. The price goes up and down, respectively.
Now what happens if k ≠ 1? Well, you get something that looks more like this:

In this case the derivative dD/dS is changing, so our price is changing. In this case we can associate D with NGDP, S with the amount of currency (M) and p = dD/dS with the price level. The rate of change of the price level is inflation

Also note that you can think of adding money to an economy like adding gas and making the demand for gas (the sale price) go up. In the case of money, NGDP goes up by a larger percentage than the amount of money you added, though (until you end up in a liquidity trap).

I am under the impression that many people who say they don't understand what I am talking about may be thinking that the information equilibrium approach is a lot more complicated than it really is. We are essentially broadening the definition of what it means for two things to be equal. You could start with two functions being equal, A(t) = B(t), and move from there to two functions being proportional, A(t) ~ B(t), so that A(t) = c B(t) + b. Information equilibrium goes another step so that two functions are in information equilibrium if, heuristically, log A(t) = k log B(t) + b.

If you have two functions related by log A(t) = k log B(t) + b, then it is possible to send a message by modulating B(t) and have it show up (faithfully) in A(t) [2]. If they are not, then the message will be diminished (you could lose some SNR) or you might not be able to faithfully reconstruct the original message (e.g. ambiguities)

Why is this useful in economics? Because there are lots of times when economists want to relate two quantities (like the price level and the money supply, interest rates and NGDP, or hourly wages and the unemployment rate) in order to understand the macroeconomy. The relationship log A ~ k log B is a minimal condition that must be met, otherwise there is information loss (non-ideal information transfer) or some piece of your model that you haven't accounted for. Another way, if you can't relate A and B via log A ~ k log B, then your theory is wrong or incomplete. There is also the additional check of the equations involving the price -- you have A = k p B or more generally A = k (dA/dB) B, but also log p ~ (k -1) log B (using the solution to the differential equation). I've found that things with trivial prices (where p is constant or approximately constant) are not terribly useful models.

This is an incredibly useful tool to sort out some of the bullshit ad hoc theories out there!


[1] I started writing up a few paragraphs that attempt to field some of the more obvious objections, but ended up with another Socratic dialog. It'll be in a future post.

[2] Before you ask "what about radio waves?", just take your function to be exp(i A(t)).


  1. Hi Jason,

    I like this post. I haven’t commented for a few months but, as a reminder, I’m not an economist or a physicist. My background is in business and government. I’d like to make a couple of comments.

    First, on January 27, you wrote a post where you quoted someone called Lee Smolin. Here is one of the quotes:

    “The observables of economics are accounting and other records. One should then try to construct a theory of economics that involves only observables. The importance of this kind of operational principle in physics and other sciences have been paramount”

    I agree with this 100%. In fact, I cheered out loud when I read it. This principle is paramount not only in science but in business and government and any other real-world endeavour.

    However, it’s not paramount in mainstream economics which appears to be built on a series of dubious mental ‘gadgets’. Take supply and demand curves. When Apple offers to sell us an iPad for $500, we observe the demand at that price during a specific time period. We don’t observe the demand at a price of $400 or $600. We can use the mental gadget of supply and demand curves to speculate what the demand would have been at different prices, but we can’t observe these alternative reality demands.

    In fact, we don’t necessarily observe either supply or demand at $500 either. It’s perfectly possible that there was a higher demand but Apple didn’t have access to the production capacity to meet that demand. It’s also possible that Apple manufactured additional iPads but could only find demand for the ones which were sold. Only Apple knows the actual ‘observed’ supply and demand. All that economists can observe is what was actually sold (or purchased, as actual sales always equal actual purchases).

    It’s even worse when we consider assets such as shares in Apple. An increase in price can lead to an increase in demand, while a decrease in price can lead to demand falling through the floor. How does that fit with supply and demand curves?

    What I don’t understand in your blog is why you seem to see supply and demand curves as central to macro-economics. I can understand your fuel pump example without involving supply and demand curves. Indeed, supply and demand curves seem irrelevant to me so they only confuse the message of the example. Similarly, I don’t see the relevance of supply and demand curves to observations (actually estimates rather than observations) of GDP, money supply and inflation.

    Your arguments would work better for me if you stuck rigidly to observables. There are three reasons for this. First, they would be simpler and easier to understand. Second, they would be more consistent with Smolin’s view on observables. Third, they would be more compelling for non-economists like me who are doubtful about economists’ tendency to create whole mental worlds from dubious gadgets and to ignore observations which are inconsistent with those gadgets.

    1. Hi Jamie

      I wrote up a response as a whole new post:

      I hope I answered all of your questions or addressed your arguments. If I missed anything, feel free to bring it up. I just realized I've done 400 posts. Needless to say, I enjoy talking about this stuff -- especially addressing critical takes.

      BTW, Lee Smolin is a theoretical physicist -- most well known for being the developer of loop quantum gravity (a competing theory against string theory to understand how quantum mechanics and gravity fit together).

  2. Second, mainstream economists and you seem to focus on trying to forecast the future using your favoured equations but you don’t ever talk about cause and effect between the variables in the equations. However, in business and government systems, understanding cause and effect is as central to a ‘scientific’ approach as using observables.

    Take a different fuel-related example. Supposed mainstream economists observe a car at monthly intervals. They measure distance travelled and fuel used. They can see that there is a relation between these observations. Roughly

    D = m * F

    where D is distance travelled, F is fuel used and m is the “fuel multiplier”.

    The key question for me, which economists and you don’t seem to ask, is which of F and D is the cause and which is the effect. Which of the following statements is more accurate?

    1) An increase in fuel causes the car to travel further

    2) An increase in the use of the car for travel causes an increase in the fuel.

    I’d say that the second statement is more accurate. No-one refuels their car and then asks how they might use that fuel. Rather, they decide to travel somewhere and then obtain the fuel required to make the journey.

    One of the main disputes in macro-economics is between Market Monetarists who seem to think that an increase in money causes an increase in GDP, and Keynesians who seem to think that an increase in animal spirits causes an increase in GDP which, in turn, causes an increase in the supply of money required to allow the GDP to take place. This is central to any policy response. In my fuel analogy, Market Monetarists think that the Central Fuel Provider can make the car travel further by putting more and more fuel in the tank. In fact, all that will happen is that the fuel tank will fill up and then flood the service station with the surplus fuel.

    Does the information transfer model have anything to say about cause and effect?

    Finally, I asked you some time ago if you had looked at any heterodox economics. One of the reason for this is that Post-Keynesian economists emphasise the importance of building economic theories from the observables and cause-effect relationships in the operational economy e.g. how does the banking system really work. If you read Nick Rowe’s blog, he often refers to “people of the concrete steppes”. Many of the people he’s referring to think of themselves as Post-Keynesians. Also, Post-Keynesians often say things like “Paul Krugman is right but for the wrong reasons”. I have seen you saying similar things.

    1. Sorry Jamie, my original comment here failed to show up for some reason.

      I would say that the information transfer model lives on the concrete steppes and that I said exactly that about Paul Krugman in this post:

      Regarding the post-Keynesian economics, it seems to be a broad umbrella and the only thing in common is that they reject the neo-Keynesian comprimise with the neoclassicals and the economy won't always return to a 'natural rate' of unemployment per Keynes General Theory.

      If you have any good recommendations of blogs to follow to understand what it is about a little better, that would be great.

      For awhile I followed Edward Lambert at Angry Bear, who talks a lot about effective demand (another big post-Keynesian principle). He also seemed to be a neo-Fisherite, so I'm not sure where he really falls. I stopped following him when he said that he just made up a formula on his blog.

      Again, any recommendations would be welcome.

    2. Thanks for this reply and your subsequent post. I've read the post and will need to think about how to structure my reply but I will reply.

      It's probably better to come back to Post-Keynesian economics after a broader discussion. I think you are doing Post-Keynesian economists a disservice if you think they are defined by what they are against. They are, however, very bad at putting forward their case in a way that appeals to anyone apart from other Post-Keynesian economists so the world draws the same conclusion as you.

    3. Cheers, Jamie. I could re-phrase "reject the neo-Keynesian comprimise" instead as "believe in the existence of high unemployment equilibria", which is logically equivalent but isn't defined with reference to other economic 'schools' of thought. I didn't mean to offend and I think a diversity of viewpoints is lacking in macroeconomics, espeically given how uninformative the data is. Economists in general -- and subscribers to these different schools -- should probably be much more equivocal.

  3. No offense Jason... and this isn't about your post here, but I was curious to see what else you had there and I have to say... most boring youtube channel ever. ;^)


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