Chapter 9 of Steve Keen’s *Debunking Economics* criticises economist’s reliance on static models in a clearly dynamic system. He first shows both Walrasian equilibrium and Gerard Debreu’s related models to be highly questionable – this is, of course, not difficult, and will be met with ‘we have improved on those!’ However, the real message of the chapter is that static analysis is fairly worthless, and dynamic analysis does not simply ‘fill in the gaps’ between different equilibria.

If you are not familiar with Walras or Debreu, prepare to be amazed at how clearly unlike the real economy these models are.

Walrasian equilibrium supposes that an auctioneer has control over the buying and selling of every commodity, and determines the ‘market clearing’ price – where supply equals demand for every commodity – before any trade takes place. Walras suggested that the auctioneer start with a random guess, which would probably be wrong. They’d then go on to adjust prices until equilibrium was reached, at which point trade would take place.

Keen refrains from commenting substantially on the realism of this approach, instead taking his usual route of accepting economist’s logic, then showing that the model still can’t work. The maths is somewhat over my head, but Keen channels John Blatt, who uses a theorem of matrices – a mathematical system by which a Walrasian auction can be explained – to show that the auctioneers prices will not converge towards equilibrium.

Simply put, there are two conditions required to Walras’ auction to ‘work:’

- The system must be able to reproduce itself e.g. produce enough iron for the required inputs of iron in the next period.
- The prices must be ‘feasible;’ basically, they cannot be negative.

According to Blatt, these two conditions require the matrix and its inverse to have the same properties. In English, this means that something and its opposite have to have the same properties, which is obviously logically impossible. Hence, the auction will not converge to equilibrium.

Debreu did not worry about whether an economy would converge to equilibrium, but simply whether or not an equilibrium existed. However, the same conditions outlined above – not to mention the incredibly restrictive assumptions of Debreu’s model, such as virtually identical, prophetic actors – showed that even if equilibrium were achieved, it would be unstable.

Keen concludes that the elusive search for equilibrium is a dead end, and moves on to chaos theory, in which equilibria are unstable and rarely or never, reached, but clear patterns emerge:

The two ‘eyes’ here are the equilibria, and as you can see they are quite clearly not worth studying – what is instead needed are differential equations that describe the dynamic evolution of the system. Economists* do* have a more advanced definition of equilibrium, which states the time path for the economy, but it involves restrictive assumptions similar to Debreu’s, and is not on the same level of dynamism as chaos theory. Anyone untainted by neoclassicism will be able to see that the above pattern is similar in type to the cyclical behaviour of a capitalist economy, and that applying chaos theory to economics is surely an idea with potential.

Keen ends the chapter by giving a couple of examples of attempted dynamic (though not chaotic) analysis – the Goodwin model, based on Marx’s analysis of the relationship between wages, investment and capital, and A .W. Phillip’s ill-fated attempts to bring dynamic modelling into economics. Contrary to popular belief, Phillips was well aware of expectations and how they changed, and incorporated this into his model. Both of these models produced realistic business cycles, as do Keen’s similar model (which we will come to in a later post).

But economists reject this type of analysis because…engineers don’t know what they are doing? The empirically successful microfoundations project? Assumptions don’t matter but do when they aren’t ours? I honestly just don’t understand.

#1 by

ivansmlon August 14, 2012 - 10:48 pmCouple of comments:

Regarding stability of equilibrium, could you try to describe Blatt’s argument in a bit more detail? Because certainly there are plenty of properties that can be true for matrix and its inverse both (e.g. positive definiteness), so that doesn’t narrow it down very much. Anyway, GE theorists have spent some time studying equilibrium stability, although the usual approach is about prices adjusting in “fictional” time, before any trades takes places (so-called tatonnement). What they found is that one can find examples where tatonnement is unstable, and one can find also cases and conditions under which it is stable. And sure, that might be a problem, but I don’t think it proves equilibrium worthless – and it’s a bit strange if Keen doesn’t discuss this literature at all.

Also, I’m pretty sure Debreu’s model does NOT require identical actors.

Regarding chaotic dynamics, some economists have actually studied such models – see Palgrave entry by Benhabib or Boldrin & Woodford (1990) for some references. I’m no expert on this, so I can’t say why the idea didn’t stick around – but what’s more interesting here is that those chaotic models are _equilibrium_ models. People maximize utility, their plans are mutually consistent and their expectations correct, and yet the resulting dynamics is chaotic. Thus, the dilemma between equilibrium and chaos, presented by Keen, is simply false.

In fact, it seems to me that Keen is making precisely the same mistake you were warning about, i.e. confusing equilibrium as static, steady state and equilibrium as a whole time path (or with uncertainty, stochastic process). Yet he gets a pass?

#2 by Unlearningecon on August 15, 2012 - 5:24 pm

As I said the maths is a bit over my head and I have not read Blatt myself, but from what I gather he uses the Leontif input-output matrix to compute the characteristics of Walrasian equilibrium, and finds that the matrix and its inverse must have the same properties for the prices to be feasible and the system to be able to reproduce itself.

Walras and Debreu are, AFAIK, fixed point equilibriums rather than time path equilibriums. Economists use both definitions, and time path equilibrium is not the same as chaos anyway.

I have only skimmed those papers but they don’t even seem to use differential equations, and retain standard neoclassical concepts such as utility. Also, calling something ‘chaotic’ doesn’t mean it is chaos theory. Having said that, there have been attempts to incorporate chaos theory into economics, yet they haven’t taken off because they don’t fit the standard story.

Debreu’s models do not require identical actors; I said ‘virtually identical.’

#3 by

ivansmlon August 15, 2012 - 9:54 pmYes, original Arrow-Debreu model is static, but you can expand the commodity space to include dated goods and you have time-path equilibrium. My point was that Keen seems to understands equilibrium as single point (the eye in Lorenz attractor), which we both agree is not the way equilibrium is understood in mainstream economics (in models with time dimension).

The papers I cited show mostly discrete-time models – so what? Chaos can exist and can be rigorously defined in both discrete and continuous time (though I’m also not very familiar with this type of math, B&W paper has some references to more technical literature). That they retain neoclassical concepts is pretty much my point – if you criticize neoclassical economists for refusing to consider models with complicated endogenous dynamics, when in fact they have considered such models, your critique is simply wrong.

A much more interesting discussion would be about actual merits of different approaches – models with deterministic, and potentially complicated dynamics, vs. models with simple dynamics and stochastic shocks. For example, Woodford focused on the first type at the beginning of his career, but then switched to studying monetary policy in New-Keynesian models (which are of the second type), so it would be interesting to know what changed his mind.

#4 by

Minon August 17, 2012 - 11:32 amivansml: “Thus, the dilemma between equilibrium and chaos, presented by Keen, is simply false.”

The idea of an equilibrium is not simply that certain things are equal, it is that states close to the equilibrium converge to it or oscillate around it, remaining close to it. It takes significant perturbations to escape from equilibria.

By contrast, chaos is characterized by the fact that small changes can have large effects. This is dramatized by the term, the Butterfly Effect, where supposedly the flapping of a butterfly’s wings can cause significant atmospheric turbulence.

It is true that chaos and stability can go together. For instance, the solar system is chaotic. However, it is chaotic only on a time scale of hundreds of thousands of years. On the scale of a human lifetime, we may currently regard it as stable. :)

IMO, economies (indeed, all human systems) are chaotic, or are on the edge of chaos (felicitous phrase!).

#5 by

Minon August 17, 2012 - 11:34 amThus I often refer to economic pseudo-equilibria. :)

#6 by

Blue Auroraon August 15, 2012 - 1:50 amWelcome back, Unlearningecon! How was your holiday in Greece?

Also, regarding dynamics…it isn’t enough to use dynamics. One also needs proper statistical testing. John Maynard Keynes did not approve of Tinbergen’s use of the standard normal distribution when it came to investment fluctuations in the business cycle because he correctly suspected that the standard normal distribution would not fit the data. Dr. Michael Emmett Brady believes that John Maynard Keynes was actually politely asking Tinbergen and the econometricians to test the logical foundations of his statistical testing (i.e., supply a goodness of fit test to see if the techniques worked and the standard normal distribution fit). John Maynard Keynes did not object to the use of econometric techniques for the economics of consumption because he figured that the standard normal distribution would fit the data. This might explain why Steve Keen’s empirical testing of debt deflation has a very high correlation.

Also, does Steve Keen discuss the econophysicists in this chapter?

#7 by Unlearningecon on August 15, 2012 - 6:45 pm

He actually doesn’t, he focuses more on weather scientists. I will get to it, don’t worry! :)

I’m confused about the link between consumption and debt-deflation? Is debt-deflation not about assets and investment?

#8 by

Blue Auroraon August 16, 2012 - 1:31 amWell, debt deflation is about purchasing power and savings. However, what I’m saying is, goodness of fit tests suggest that the standard normal distribution fits the data for the economics of consumption. Perhaps the standard normal distribution, for similar reasons, debt.

#9 by

KevinMon August 15, 2012 - 2:52 pmSo the model doesn’t work… but in the end there is a market, and there will be winners, and the winners will assert a system, and the system will be based on something they understand. Even if what they understand is a comically inaccurate model.

#10 by Unlearningecon on August 16, 2012 - 6:47 pm

There are links between neoliberalism and big money.

Labour market economics was invented at a time of labour unrest purely to defend the status quo.

#11 by

Christianon August 16, 2012 - 10:34 pmHi, Unlearning,

I, too, distrust neoclassical economics ideology and am probably very sympathetic to your political views. I found your blog some weeks ago, and have really enjoyed following it so far. However as a math PhD student, I have to agree, that it is certainly possible for both a(n) (invertible) matrix and its inverse to have the same properties. This really depends on the properties under consideration. I also disagree with your ‘translation’ of the property into english (‘In English, this means that something and its opposite have to have the same properties’). The word opposite is much to vague and ill defined for one to be able to say that the inverse of a matrix is its ‘opposite’. Also, the vagueness and illdefinedness of the word ‘opposite’ make it hard to conclude that the statement of the english ‘translation’ is ‘impossible’. Clearly a reasonable person would agree that, in a certain sense, south is the opposite of north, but both have the property of describing cardinal directions.

In solidarity,

Christian

#12 by Unlearningecon on August 17, 2012 - 4:28 pm

Yes, you are absolutely right and I was waiting for somebody to call me out on this. I just wanted a way to communicate it to everybody – perhaps I oversimplified to the point of misleading.

Of course, the property he identifies – though I cannot seem to find what it is – will be a property that a matrix and its inverse cannot both have, such as the determinant being greater than 1.

Thanks for the kind words about the blog. I have not read Myatt and Hill, but I’d go with Keen over Varoufakis, as he’s more accessible and readable. Varoufakis is probably

better, though, in the sense of attention to detail. Perhaps you’d like to read that afterwards. (Btw, I get the impression Myatt and Hill is the most accessible of the 3).#13 by

Christianon August 16, 2012 - 10:47 pmBTW, when I said that I have ‘enjoyed following your blog so far’ this is really an understatement. I think it is fantastic that you are doing the work of allowing those with a background in economics or the social sciences such as myself to learn what is wrong with economics (or at least to learn that is neither scientific nor value-free, but very ideological).

Between Varoufakis books, Myatt and Hills book and Keen’s book, which one do you recommend? I am sure that a reading guide for the interested layperson would be much appreciated.

Cheers,

Christian (Could you change my name in my previous posts? CENSORED is not a very common name and it is probably easy to track down the few math graduate students with my first name. I am a bit paranoid about this sort of thing)

#14 by

Minon August 17, 2012 - 11:49 am“Simply put, there are two conditions required to Walras’ auction to ‘work:’

“The system must be able to reproduce itself e.g. produce enough iron for the required inputs of iron in the next period.

“The prices must be ‘feasible;’ basically, they cannot be negative.

“According to Blatt, these two conditions require the matrix and its inverse to have the same properties. In English, this means that something and its opposite have to have the same properties, which is obviously logically impossible. Hence, the auction will not converge to equilibrium.”

No, that is not what it means. Suppose that you are in an equilibrium. Then, no matter what the direction of time is, you remain in equilibrium. That is not a logical impossibility.

In real life, you have processes that diverge, that oscillate, or that converge. (Take that, Nick Rowe! ;)) You also have random changes (noise), perturbations, and other shocks. That means that in practice we do not experience true equilibria. Now, if you have a process that converges to an equilibrium or oscillates around one, it helps to know what the equilibrium is. That helps us to understand what is going on, even if we do not actually reach the equilibrium. :)

#15 by

ivansmlon August 17, 2012 - 1:38 pmAgain, you are applying physicists’ concept of equilibrium, which is different from economists’ concept. In economics, equilibrium means set of prices and quantities such that 1) agents would choose such quantities, if they optimized their utility/profits given such prices, and 2) quantities are mutually compatible, i.e. markets clear, i.e. total supply = total demand, for each good. Set of prices and quantities can include multiple time periods, and equilibrium does not require that economy stays at one static state through all those periods. It is perfectly possible that economic variables fluctuate, there are shocks to the economy, or economy grows over time and yet, both 1) and 2) are satisfied at all points in time.

What distinguishes equilibrium from disequilibrium is that in the second case, plans of individuals will not be mutually consistent. There will be excess demand or supply, and some plans will not be fulfilled. Such situation is of course possible, and whether the economy adjusts from disequilibrium to equilibrium is an interesting question. Even more interesting question is which one is more appropriate to describe particular situations – if a business accumulates inventory, is it because of excess supply, or is it optimal equilibrium response to changed expectations of future demand? But all this straw-man equilibrium-bashing brings exactly zero insight to these topics.

#16 by

Minon August 17, 2012 - 7:10 pmJust to clarify: My view of equilibria is that of game theory and systems in general, human or otherwise. :)

#17 by Unlearningecon on August 17, 2012 - 5:01 pm

Ivan,

Yes, it would have been nice for Keen for address the second economic type of equilibrium.

What I’m, getting at is that even this definition of equilibrium effectively means that certain characteristics – preferences, behaviour – are in equilibrium over a period of time. A chaos theory-esque approach would be built up specifically from the changing nature of these things – or, more likely, more aggregative macroeconomic variables.

Min,

I believe I oversimplified somewhat – it would really help if I had access to Blatt. From what Keen says he does show that the system will not converge to equilibrium, and that equilibrium is unstable. But I’ll have to take Keen’s word for it this time.

#18 by

pontuson August 18, 2012 - 12:07 pmI am personally leaning quite favorably to the neoclassical way of thinking. Take, for instance, Brian Arthur’s El Farol Bar Problem (http://tuvalu.santafe.edu/~wbarthur/Papers/EconandComplex_Web.pdf)

As Arthur notes, there is no pure strategy equilibrium in this game. But there is a mixed strategy equilibrium; that each agent with 60% probability goes to the bar, and with 40% probability stays at home. The resulting outcome is a participation rate of, of course, 60%. This is the neoclassical view.

Now, using Arthur’s approach is really fascinating. The equilibrium displays tremendous complexity and is, in fact, never in “equilibrium”. Yet, it oscillates very closely around the equilibrium predicted by “neoclassical” theory.

So is the fact that Arthur’s economy (if we treat that as the “real world”) never is in equilibrium a failure for neoclassical theory? I think not. With a few equations, neoclassical theory predicted an outcome which could easily be understood. The alternative was computational crunching for days after which a black-box result emerged. Even if the latter is more “realistic”, or perhaps even slightly more accurate, I am not sure if accuracy should triumph insight when we try to understand the world.

#19 by Unlearningecon on August 18, 2012 - 4:13 pm

I don’t know as much about game theory as I do about marginalism – it could well be said that my main beef is with marginalism. There are many areas of economics that are less rigid and more empirics based.

That article is interesting, but I don’t think it’s particularly ‘neoclassical;’ it seems to diverge knowingly from the common approach.

I don’t necessarily think that equilibrium is unstable is a failure of neoclassical theory – the point here was that Walras believed his auctioneer process would converge to equilibrium, and Debreu devoted his time to seeing if an equilibrium exists. But results like that paper, or chaos theory, reveal that these are dead ends, and dynamic paths are a better area for study.

#20 by

Minon August 17, 2012 - 6:50 pmUnlearning: “What I’m, getting at is that even this definition of equilibrium effectively means that certain characteristics – preferences, behaviour – are in equilibrium over a period of time.”

Otherwise, why call it an equilibrium? :)

#21 by

Roman P.on August 17, 2012 - 10:42 pmI think that some of the older neoclassical economists did actually believe in a static-state equilibrium and so this way of thinking propagated amongst their students. Knight (1924), for example, admits that in his theories economic interactions are static and dynamics only enters in the economics of development. And the modern ones… Dunno, don’t learn ODE?

#22 by Unlearningecon on August 18, 2012 - 2:35 pm

Yes, absolutely. The new definition of equilibrium is quite modern, and unique to DSGE. Other models generally mean a static state equilibrium (Walras, demand-supply).

#23 by

vimothyon September 3, 2012 - 1:22 pmPapers that study economies with chaotic dynamics:

Bohm & Kaas (2000), Benhabib & Nishimura (1985), Boldrin & Montrucchio, (1986), Sorger (1992), Nishimura et al (1994), Venditti (1998), Mitra and Sorger (1999), Brock & Homes (1998), Chiarella (1988), Barnett & Serletis (2000), etc, etc, etc…

#24 by Unlearningecon on September 3, 2012 - 3:22 pm

First, a summary of a couple rather than a list of names would have been more constructive.

Second, as I pointed out above, calling your paper ‘chaotic’ is not the same as it being chaos theory. The papers I looked at are still DSGE equilibrium models, with all the standard representative agents maximising utility etc. This isn’t what I, Keen or the Econophysicists are suggesting.

#25 by

vimothyon September 3, 2012 - 4:19 pmMany of the papers above feature representative agents in infinite horizon models. The first is basically a modification of Solow-Swan.

“Chaos theory” is the study of chaotic dynamics. All of the papers I cited feature chaotic dynamics, so I’m not sure what point you’re trying to make there.

#26 by Unlearningecon on September 3, 2012 - 10:45 pm

Chaos theory is the study of deterministic systems whose behaviour is strongly reliant on their initial conditions, and unpredictable, despite not necessarily being random. They are modeled with differential equations.

This is completely the opposite of DSGE models, even if the latter can be tweaked to create ‘chaotic’ dynamics.

#27 by

ivansmlon September 3, 2012 - 11:51 pm(if I may)

1) as I already wrote, chaotic models can be modelled with either differential or difference equations. The canonical example of chaos, logistic map x’ = 4*x*(1-x) (prime denotes next period), is cast in discrete time,

and

2) fortunately, what “chaotic” means doesn’t depend on me, vimothy, you, Keen, econophysicists or neoclassical economists, but on mathematicians, who have all the precise definitions and theorems. Given a discrete-time dynamical system x’ = f(x), where x is vector of all relevant variables, or continuous-time system dx/dt = f(x), the question whether it exhibits chaotic dynamics depends only on the properties of function f. This has absolutely nothing to do with original economic assumptions from which f was derived. And as those cited papers show, it is possible to derive f with chaotic dynamics from neoclassical assumptions. So unless you can find mathematical mistakes in those papers, I really don’t understand your objection.

#28 by Unlearningecon on September 4, 2012 - 5:30 pm

Fair point about difference equations, but all I’m saying is that economics does not use techniques that are analogous to chaos theory. You can call the behaviour of certain DSGE models ‘chaotic,’ but it does not have the same parameters and methodology as what is commonly considered chaos theory.