Wednesday, February 3, 2016

Attainable definitions of equilibrium

An article from Steve Keen was reprinted at Evonomics that makes some pretty bold claims. One of them is effectively a restatement of an old result about the stability of  t√Ętonnement given some assumptions of how it works:
... the original “Neoclassical” mathematical model of a market economy is mathematically unstable: it doesn’t converge to a stable pattern of relative prices and a stable growth path for the economy ... the argument ... was first made in the 1960s by Jorgenson (who was applying a mathematical theorem from the early 1900s) ...
There are a lot of mentions of math at the beginning.

I went back to Keen's lecture (linked in the article) to get the details of the argument, which is in general sensible. The matrices that evolve output and prices from one period to the next are proportional to R and R⁻¹. The stability of iterating R and R⁻¹ depends on their eigenvalues. But R and R⁻¹ are non-negative so one of R and R⁻¹ is going to lead to an "unstable" (non-equilibrium) result ... the growth rate of some good (its output or price) will diverge from the "equilibrium growth rate" (of output or prices).

As I said, makes sense. But it hinges critically on the definition of equilibrium. This definition (from Keen's lecture):

– [Two] conditions apply for a growing economy to be in equilibrium:

  1. Output of every good must be growing at the same rate
  2. Relative prices must be constant

Well, then an economy isn't in equilibrium by definition because this isn't true of e.g. the US economy. At least this is consistent: if an economy was in equilibrium, then it couldn't be growing, but since it's growing, it's not in equilibrium.

But we don't need the fancy proof. If equilibrium is defined as it is above, then economies are not in equilibrium, QED. Could an economy be in equilibrium with this definition? According to the proof, no. So we have a definition of equilibrium where not only is the observed state not in equilibrium, there is no chance of any observed state ever being in equilibrium.

Is a definition that nothing can meet a useful definition?

NOPE

Anyway the actual neoclassical definition of equilibrium is that there is no excess supply or demand [1]. That's what Walras was talking about with t√Ętonnement -- prices change in response to excess supply and demand in order to bring them back to zero. There is nothing in there requiring the output of each good to grow at the same rate. Computers can grow in output at 20% per year and bread at 1% with supply of computers and bread being equal to demand for computers and bread, respectively, at each time step.

And really, neoclassical equilibrium is a pretty decent model. We don't show up to the grocery store with piles of rutabegas one day, piles of cheddar then next, or empty places where the bacon should be most of the time. That is to a first order approximation roughly zero excess demand and zero excess supply. I made an economics joke graph:


The neoclassical model of equilibrium unemployment says that the employment rate e ≈ 100%. The real world actually approximates this (at least in the post-war US) fairly well with employment varying from e ≈ 85% to e ≈ 96%. The later natural rate and matching models predict something closer to e ≈ 94-95% most of the time -- a bit better.

I wrote in the past that mathematical theorems are usually not a good fit for real human systems. If someone has claimed a "mathematical proof" that the economy is unstable (or stable), you have to take it with a grain of salt. The real world likely only approximates the strict mathematical conditions for the proof of stability or instability, so you can at best say the real world is "approximately stable" or "approximately unstable".

As always, I must bring the discussion back to the information equilibrium model. In the partition function approach (discussed in the paper), an economy has a picture that looks like this:


Both prices and nominal output are made up of "growth states" (think inflation for prices) that look like this (typically, the distribution for prices -- inflation rates -- has a lower average than the one for nominal output -- hence real growth is positive). Equilibrium is the state where there is no change in the distribution -- and it is defined by the average growth rate. However individual prices and outputs can move from one state to another. An analogy is a Maxwell distribution of velocities in an ideal gas. The distribution (in equilibrium) doesn't change, but individual atoms can scatter off each other, resulting in changes in their velocities.

That means Keen's definition of equilibrium doesn't apply. Relative prices are changing all the time, and output growth for every good is different (and changing).

It also looks more like a real world.

And at least it's a definition of equilibrium that isn't unattainable.

...


Footnotes

[1] Keen's critique of the method of restoring equilibrium is more applicable to the Ramsey-Kass-Coopmans (neoclassical growth) model. I talk about that instability here. The "people will see the problem and correct it" argument makes the entire model about people seeing the problem and correcting it, and not about the RCK equations at all. But in RCK model there is simply an aggregate "consumption" -- so there aren't outputs for different goods that could have differential growth.

7 comments:

  1. I listened to Keen's lecture and I had the impression that this definition of equilibrium wasn't something he invented, but came from somebody else. Jorgenson perhaps?

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    1. It is called "balanced-growth equilibrium" (I think it originates with Samuelson and Solow and neoclassical growth models), but it's not really what Walras had in mind. And you can't just go around saying it's "equilibrium" ... or even "neoclassical equilibrium".

      https://en.wikipedia.org/wiki/Balanced-growth_equilibrium

      The main reason for invoking it that the things that don't grow at balanced growth rate g, at infinity, are either the entire economy (if g' > g) or are negligible (g' < g).

      Keen says the economy is unstable because it is impossible to follow the balanced growth equilibrium. People not familiar with economics would probably take this and say: Look, neoclassical growth isn't stable! It's going to eventually collapse after a series of boom/bust cycles!

      And my problem is that I think Keen wants people to come away with that impression.

      But that's not what unbalanced growth means. It just means that either you have a set of things with an average growth rate and the composition of your economy is changing. Auto companies are the fast growing sector in one decade, tech companies are the high growth industry at another time, financial companies another, and maybe eventually health care.

      Sound familiar?

      No economy actually has all of its firms growing at the same rate. That's a ludicrous standard for "equilibrium". It simplifies the discussion of growth models at infinity, though.

      But economists don't handle infinity very well. Basically when you say there is one growth rate g, that is one time scale 1/g and there are only two meaningful periods in your economy: T << 1/g ("zero") and T >> 1/g ("infinity").

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  2. "And my problem is that I think Keen wants people to come away with that impression."

    Yes, unfortunately I also get that sense from him.

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  3. I have a different take on Keen. To me it seems pretty clear that his ideas fall into a tradition in the social and life sciences about complex systems and chaos theory. Some complex systems do not have stable equilibria, but they are nonetheless self organizing. If the economy is such a system, it is not, as economics dogma holds, self correcting. In that view, Greenspan's (The Maestro's) claim that there is no need to have or enforce laws against financial fraud, because the financial markets are self correcting is obvious bullshit. And, as we now know, massive financial fraud, from liar's loans to fraudulent credit ratings, was a large part of what caused the recent financial crisis. Hoover's Treasury Secretary Mellon also believed that the economy would quickly self correct after purging its weaknesses, as it had in 1873. That was also a disastrous miscalculation.

    Also, if the economy is self organizing but not self correcting, then it contains within itself the seeds of its own instability. To followers of Minsky the socalled Great Moderation did not mean the end of depressions, but signaled that people would take on too much risk, unaware of the dangers. That proved to be the case, despite the warning signals of the S&L crisis and the Long Term Capital Management debacle.

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    1. Bill, in light of what you say about Keen & Minsky (and assuming it's true) it makes me wonder if there's something to be said for purposely causing small recessions on occasion: like setting off dynamite on snowy slopes to reduce the chance of a big unplanned avalanche. Or doing a "controlled burn" to eliminate fuel for an unplanned and uncontrolled conflagration. I'll talk it over with the rest of the Trilateral Commission and see if there's interest in pursuing this idea further. :D

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    2. CBs are already causing recessions on purpose. The economy pretty predictably has a quick rebound when they take their foot off the brake. The slow recoveries from the last couple of US recessions are clues that they were not caused on purpose by the Fed, but had other causes, even if the Fed contributed. These slow rebounds suggest to me that CBs can easily slow economies down, but cannot easily speed them up.

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