Sunday, July 13, 2014

In defense of equilibrium

I signed up for the comment updates at John Quiggin's post and they have been flooding in for the past couple days. I would like to say that the comments are very thoughtful and erudite (i.e. spelled correctly). Crooked Timber seems to have that effect. One thread that was pervasive was discussion of equilibrium, in particular a reference to Krugman's blog post where he describes "useful economics":
"So how do you do useful economics? In general, what we really do is combine maximization-and-equilibrium as a first cut with a variety of ad hoc modifications reflecting what seem to be empirical regularities about how both individual behavior and markets depart from this idealized case. And people using this kind of rough-and-ready approach have done really well since 2008, on everything from inflation to interest rates to the effects of austerity."
Now the definition Krugman is using here is effectively the "ADM" equilibrium -- there exists a set of prices so that excess demand is zero (or, there exist a set of prices that clear the market). It is important to keep that definition in your head though the rest of this post, because many comments seemed to see this as some sort of shocking revelation that the entire edifice of economics was based on a massive logic fail. For example:
"As an operational model, maximization-and-equilibrium isn’t ideal — it doesn’t make any sense at all. In a world of radical uncertainty, where people do not know what they do not know, 'maximization' can not defined. 'Equilibrium' is even more vacuous — in practice, it comes down to what pedants call, the ergodic hypothesis, a crazy idea that history doesn’t matter — particularly when you consider that capitalism is one strategic investment after another in making history matter."
Strategic investment is "maximization", so I'm not sure what that's all about. Plus the other side of that "investment" is losing out and so there's your egodicity. Snark aside, in the past history has not seemed to matter -- at least with regard to economic variables. The economy appears to return to a long run trend (Milton Friedman's plucking model). This has been a major debate: does RGDP growth have unit root? The argument is inconclusive based on the data alone and you need to assume and underlying economic model in order to say whether the ergodic hypothesis holds or fails. You can't just claim ergodicity is crazy without specifying a model. In the information transfer model, the "ergodic hypothesis" seems to hold. That is evidence that if ergodicity appears to be crazy in your model, then your model is likely crazy.

These are the comments that set me off though, pushing me to become the non-economist champion of the besieged economists:
"By hammering at equilibrium no matter how often it fails to map our experience with the world, academic orthodox Econ simply refuses to notice a problem."
"... equilibrium (demand equals supply) is probably not a good way to understand a dynamic, chaotic, system characterized by violent reversals."
These comments seem to suffer from availability heuristic -- only the newsworthy economic events seemed are the valid subject of scientific economic inquiry.

And in their defense, there aren't many news stories on Bloomberg where supply met demand through the price mechanism yet again. Even the massive failure of macroeconomic management that was the great recession seemed to only shave about 3.5% off of NGDP. That's right -- about 96.5% of transaction value (more than 14 trillion dollars) was basically unaffected. Bloomberg disproportionately fixated on the 3.5%; so do we all.

And how many violent reversals have you seen in the price of bacon? And food is one of the more volatile prices out there -- hence why "core CPI" leaves out food.

The attacks on economists only get more confused:
"One of the worst ideas ever is that of economic equilibrium to which things return. Better would be a sense of history and a realistic view of the world we live in. Exponential growth cannot continue forever. But capitalist economies have relied on that from the beginning."
This elides the definition of equilibrium -- how do you "return" to an equilibrium in the presence of exponential growth? That exponential growth is the "equilibrium", and if exponential growth halts [1], that will be the new equilibrium, so that's a non sequitur.

I actually have a related discussion here. Before you jump on the fact that I say that equilibrium does not exist, note that I am referring to a specific equilibrium: constant inflation and constant interest rate over the long run.

Anyway, an equilibrium in the economics sense is not some status quo to which things always return. It is (remember the definition) a set of prices where aggregate supply meets aggregate demand, i.e. clear the market. I'd say that's pretty realistic view of the world seen with a sense of history: I don't see a lot of excess demands in the US (except traffic, which is a failure of the price mechanism on the scarce supply of roads at a given time) and there aren't a lot of excess supplies (except maybe in some cases of government subsidies for corn and wheat).
"The more I read about equilibrium analysis the more I couldn’t believe that anyone would begin to imagine it might say something interesting about human behavior. What was worse, I found out that in practice, nobody could point to an example of a equilibrium model that matched the data."
This is where the madness creeps in. All I could say to this is: Really? Does this commenter go out to the supermarket thinking the price of bacon will suddenly be 30 times as much one day and half as much the next, with bacon filling the entire grocery store? We operate every day with prices that are roughly the same as yesterday's and that most products you bought yesterday will be on the shelves again today. The whole concept of a price system without massive surplus or shortages, e.g. a supermarket, is a prediction of equilibrium! I should be careful. Maybe that commenter lives in Somalia and is trying to use bitcoin.

Sure there is a trend towards higher prices (inflation), but that is not inconsistent with equilibrium -- if the trend is set by monetary policy, then that trend represents equilibrium and prices return to that trend assuming there is not some massive supply or demand shock.

Another (snarky) way to put it is that a naive market equilibrium employment rate is predicted to be 100%. The current value? 93.7%. Here is a graph of the naive equilibrium model and the slightly improved natural rate model:


Economists don't tend trot this plot out (maybe they should?) because they're more interested in the deviations from equilibrium. It's actually rather amazing that the model should work this well!

That brings us finally to this comment:
"Back in the day, to derive an equilibrium analysis, you had to find a constraint, such as a conservation law."
While I definitely sympathize with this approach (see e.g. here), this is a very narrow definition of equilibrium. For example, this leaves out ecological equilibria (maybe there is some as yet undiscovered conservation law?) or the global climate thermal equilibrium (which is actually a product of a non-equilibrium system).

The scientific way to go about an equilibrium analysis in this framework is first to observe what appears to be an equilibrium in the data. You then say as a scientist: maybe this is an equilibrium, and here is the constraint. This "constraint" is a statement of an empirical regularity. For example, the Phillips curve represents a loose empirical regularity between the inflation rate and the unemployment rate. The equilibrium model is then that shifts in unemployment cause shifts in inflation.

Sure the Phillips curve seemed to break down forming the basis of the Lucas critique (that says, in this framework, maybe empirical regularities aren't the way to go about discovering those constraints), but as Paul Krugman points out -- this process of identifying a constraint and deriving and equilibrium analysis is exactly how Krugman says economists go about doing useful economics.

[1] Noah Smith has an excellent takedown of a physicist (shame!) who I am a a single degree of separation from (more shame!) making the same clumsy claim as the commenter, that exponential growth has to stop.

16 comments:

  1. Have you compleated more than 4 lab classes in quantitative science or engineering courses?

    Have you compleated any degrees in physics? If so, which ones (BS,MS,or Piled Higher and Deeper)?

    What about economics?


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    Replies
    1. I have a phd in physics (I'm a theoretical particle physicist by training -- quantum field theory, string theory and such, but now I'm more of a rocket scientist -- I've stood at the top of an Atlas V rocket prepped for launch).

      I'll happily admit I have no training in economics :)

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    2. 1. What is the ODE equation of the movement of a mass connected to a spring. Virticle or horizontal your choice. You may neglect friction forces if you like. (Also, the spring is streached out such that its loops do not ever touch in compression.)

      ODE, ordinary differential equation?

      What is one solution to the ODE, and the initial conditions?


      2. What is the vector ODE equation for a satelite orbiting the earth. What would be the initial conditions. No solution requested?

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    3. Is this some kind of test? Hilarious. How about you read my thesis:

      http://arxiv.org/abs/nucl-th/0508036

      If you can remotely tell me what it is about, I'll answer your questions :)

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    4. Yes, it is a test. The very basics!

      They should be quite easy and quickly for you to answered.

      In addition, they are two examples of dynamic (non-equilibrium) situations most people can picture.

      Basics with broader applications to economics. (I must admit that I am having a hard time understanding the information theory stuff.)




      Delete
    5. Should read

      "They should be quite easy for you to answer quickly."

      Delete
    6. Come on. Its easy. Please do not deflect the question.

      Hint F(x)=.... or F(R)=....

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    7. You are incorrect in referring to a harmonic oscillator being a "non-equilibrium" situation. Dynamic is not the opposite of equilibrium; dynamic is the opposite of static. Both a stable orbit and a harmonic oscillator represent equilibrium solutions in the sense of the post above ... their time average of the energy is constant.

      This will actually help you a lot:

      H = N + 1/2

      |α> = Σ |n>

      |α(t)> = exp(-iHt)|α(0)>

      =

      = = (n + 1/2)

      i.e. the energy of a harmonic oscillator is constant, which means it is an equilibrium in the sense of equilibrium we are talking about in the post above.

      For the second question:

      L = p^2/2m - M m/q

      d/dt ∂L/∂pi - ∂L/∂qi = 0

      But again the solution for a stable orbit is an equilibrium solution (the time average of its energy is constant q.v. the virial theorem).

      The most familiar non-equilibrium situation in the sense of the post above is a pot of boiling water.

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    8. non-HTML HTML fail

      | α > = Σ | n > < n | α >

      < n | exp(iH*t) H exp(-iHt) | n > = < n | H | n > = (n + 1/2) < n | n >

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    9. This would actually you alot:

      I would have more simply given:

      1. F(X)= X"(t)M= -KX(t) as the ODE, (where X" is d/dt d/dt X.)
      K as spring constant, M is mass.


      X(0)=a ,X´(0)=0 ,as the initial condions of position and inital velocity=0.
      X(t)=a sin( sqrt(K/M)*t ) ,and the solution where.


      I am totally snowed probably as intended for your answer for number 1. I suspect it has nothing to do with a physical elastic spring?

      exp(-iHt) term looks like it might be mathematical oscillation of some sort. I do not understand the notation.

      2. I would have given this, Your equation would have been able to give this one with a G in it.

      F(R)= M R"=-(GMm)/Mag(R)¨2 UnitVector(R)
      The vector initial conditions would be R(t0) and R´(t0).

      A nifty application for your more general equation is used in structures under a different name.

      "You are incorrect in referring to a harmonic oscillator being a "non-equilibrium" situation. Dynamic is not the opposite of equilibrium; dynamic is the opposite of static."

      I was using a more basic and common (high school) definition of "mechanical equilibrium." More like static momentum (and in this case velocity ). Which the oscilator would not be in equilibrium while oscilating and the satelite would not be for eliptical orbits even though the orbit would be stable. ( I had a math teacher tell the class there are many definitions of equilibrium.) It is interesting to learn of a more theoretical definition of "equilibrium". But that definition does not negate the other one.

      http://en.wikipedia.org/wiki/Mechanical_equilibrium

      "Both a stable orbit and a harmonic oscillator represent equilibrium solutions "

      When I googled "equilibrium solutions" I got some thing I was more familiar with. Such as in the first figure. In this case the equilibrium solutions are static.

      http://tutorial.math.lamar.edu/Classes/DE/EquilibriumSolutions.aspx Text search for "equilibrium solution". This equilibrium is arrived at by doing the dynamic math, not assuming the equilibrium first.

      "But again the solution for a stable orbit is an equilibrium solution (the time average of its energy is constant q.v. the virial theorem)." That a new and interesting theorem to me but an average over time looses the dynamics in time.

      "The most familiar non-equilibrium situation in the sense of the post above is a pot of boiling water." That is a very good contrasting example for more theoretically abstract definition of (non)equilibruium you have used.

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    10. I suggest you read this:

      http://noahpinionblog.blogspot.com/2013/02/is-business-cycle-cycle.html

      Equilibrium in economics has nothing to do with equilibrium in mechanics. (If it was, it would be more well defined!)

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    11. Have you seen very much economics done with dynamic math? Math that can produce cycles with out external driving functions?

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    12. Again, I would refer you to this post

      http://noahpinionblog.blogspot.com/2013/02/is-business-cycle-cycle.html

      But yes. For example DSGE stands for "dynamic stochastic general equilibrium", which is one of the major ways economist tackle economic systems. See for example here:

      http://www.federalreserve.gov/pubs/feds/2010/201026/

      Even older models like P* are "dynamic math" as you put it:

      http://www.oecd.org/eco/outlook/34254867.pdf

      Do you honestly think that economists didn't know or somehow just forgot about basic calculus from the 1700s?

      This reference starts from stochastic differential equations that are somewhat more recent, dating from 1900:

      http://epubs.siam.org/doi/pdf/10.1137/1015001

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  2. I do do not know why you defend economists and equilbriom rather than pushing dynamics.

    Do you not want them on your more fertile lawn?

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    Replies
    1. I defend equilibrium because when I go to the supermarket the price of my favorite Irish whiskey is about 25 dollars each and every week :)

      That's equilibrium! *hic*

      Delete
  3. in 1 the verbage should have been

    X(t)=a sin( sqrt(K/M)*t ) ,as the solution.

    ReplyDelete

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