Wednesday, July 28, 2010

Equilibrium Analysis

In a recent post on his (consistently interesting) blog, David Murphy questions the value of equilibrium analysis in economics and finance, and points to two earlier posts of his in which the same point is made. Here he is in July 2007:
An interesting post on the Street Light Blog, on currency misalignments, suggests an interesting question: is economics an equilibrium discipline? The very idea of a misaligned FX rate suggests that the natural state is an aligned one: perhaps the fundamentals move faster than the markets adjust, so FX is never in equilibrium. Perhaps (in the language of statistical mechanics) the relaxation time is much longer than the average time between forcings. 
And here, in August 2008:
My own view is that finance is not an equilibrium discipline, mostly, so while classical economics might work well in explaining the price of coffee... it does rather less well in asset allocation or explaining the return distribution of financial assets. Rather new news arrives faster than the market can restore equilibrium after the last perturbation, meaning that most of the time equilibrium is not a useful concept.
In a 1975 paper that remains worth reading to this day, James Tobin was explicit about the limitations of equilibrium analysis in understanding large scale economic fluctuations:
Keynes's General Theory attempted to prove the existence of equilibrium with involuntary unemployment, and this pretension touched off a long theoretical controversy. A. C. Pigou, in particular, argued effectively that there could not be a long-run equilibrium with excess supply of labor. The predominant verdict of history is that, as a matter of pure theory, Keynes failed to prove his case.

Very likely Keynes chose the wrong battleground. Equilibrium analysis and comparative statics were the tools to which he naturally turned to express his ideas, but they were probably not the best tools for his purpose... The real issue is not the existence of a long-run static equilibrium with unemployment, but the possibility of protracted unemployment which the natural adjustments of a market economy remedy very slowly if at all. So what if, within the recherché rules of the contest, Keynes failed to establish an "underemployment equilibrium"? The phenomena he described are better regarded as disequilibrium dynamics.
Tobin then goes on to develop a dynamic disequilibrium model of the macroeconomy (discussed at length here) which has a unique equilibrium characterized by full employment, steady inflation, and correct expectations. He shows that even if this equilibrium is locally stable, so that small perturbations are self-correcting, it need not be globally stable: sufficiently large shocks to the economy can result in cumulative divergence away from equilibrium unless arrested by a significant policy response. This seems to describe what we have experienced over the past couple of years better than any equilibrium model of which I am aware.
Note that Tobin's model is deterministic. The problem here is not that the economy is being buffeted by frequent shocks that arrive before a transition to equilibrium can occur, it is that the internal dynamics of adjustment simply do not approach the equilibrium from certain (large) sets of initial states even in the absence of shocks. The idea that the instability of steady growth with respect to disequilibrium dynamics is an important feature of modern market economies, and cannot be neglected in a comprehensive theory of economic fluctuations was forcefully advanced by Richard Goodwin as far back as 1951, and Paul Samuelson had explored the possibility even earlier. As Willem Buiter has recently lamented, this line of research in macroeconomics simply dried up about a generation ago.
Another area in which equilibrium analysis is likely to be inadequate is in the study of asset markets with significant speculative activity. Price and volume dynamics in such markets depend not just on changes in fundamentals but also on the distribution of trading strategies, and this in turn adjusts under pressure of differential performance. The idea of an equilibrium composition of trading strategies is a contradiction in terms: if there were any such thing there would be a new strategy that could enter to exploit the resulting regularity. It is the complexity of this disequilibrium process that allows information arbitrage efficiency to be approximately satisfied, while allowing for significant departures from fundamental valuation efficiency (the distinction, naturally, is also due to Tobin.)
Finally consider Hyman Minsky's financial instability hypothesis, built on the paradoxical idea that stability itself can be destabilizing. In Minsky's framework stable expansions give rise to increasingly aggressive financial practices as those firms having the greatest maturity mismatch between assets and liabilities profit relative to their closest competitors. The resulting erosion in margins of safety increases financial fragility, interpreted as the likelihood that a major default will trigger a crisis of liquidity. Such a crisis eventually materializes, devastating precisely those firms whose actions gave rise to greater fragility. The balance of financial practices is then shifted in favor of increased prudence, and the stage is set for another period of stability. Trying to give this analysis an equilibrium interpretation is a futile exercise; expectations of financial market tranquility are self-falsifying, and no fixed distribution of financial practices can be stable. 
Given the potential of disequilibrium dynamic models to illuminate our understanding of the economy, why are they generally neglected in contemporary economics? In part it is because the quality of a disequilibrium model is hard to evaluate and the dynamics are necessarily arbitrary to a degree. There is a professional consensus on how equilibrium analysis should be done, but none (so far) when it comes to disequilibrium analysis. Furthermore, equilibrium models can be enormously insightful, even in applications to macroeconomics and finance. The work of John Geanakoplos on the leverage cycle is a case in point, and Abreu and Brunnermeier's paper on bubbles and crashes is another. I have used equilibrium methods frequently and will continue to do so. But it seems that there ought to be greater space in the profession for serious work on the dynamics of disequilibrium.


Update (7/31). In an email (posted with permission) David Murphy adds:
One of the main reasons people study equilibrium models is that they are an order of magnitude easier, mathematically, than non-equilibrium. If you consider a simple problem like cooling, for instance, the equilibrium version is high school physics, and the non-equilibrium version is still a research problem (with useful application to the improvement of annealing methods). Economists in my experience are very comfortable with the maths they know (a bit - stress a bit - of stochastic calculus), but they are not willing to venture much further because it gets really, really hard quite quickly.  Hence the 'we have a hammer, everything looks like a nail' problem.
I think this is correct as far as analytical results are concerned; proving theorems (aside from convergence-to-equilibrium results) with the standard toolbox is not easy in disequilibrium models.  But if one adopts a computational approach the reverse may be true. I, for one, find it easier to write an algorithm to simulate a recursive system than one that requires the computation of fixed points in high dimensional spaces. But (as noted above) there does not exist anything close to a professional consensus on how the quality of such models should be evaluated, and so they are usually rejected out of hand at most mainstream journals.


  1. This is a very interesting post. I have been following macroeconomics as an outsider (with considerable experience and background in engineering/mathematics). I still don't understand why the line of research starting from Hyman Minsky has not received more attention from the mainstream macro community. His explanation seems to make a lot of sense and probably is far more important than the endless debates between freshwater vs coast camps. Your explanation helped me a little bit in understanding this state of affairs. Where does the macro community discuss such matters? Is there any consensus on the major future research problems/directions?

    Thanks for your blog.

  2. Minsky has been very influential among practitioners since well before the crisis - see, for instance, this post by Paul McCulley of PIMCO

    Look at the website of the Levy Institute at Bard College for more information on his academic influence. There's been an annual Minsky conference for about two decades now and this has recently started to attract a lot more attention.

  3. Rajiv: instructive, as always.

    But another half of me says: hang on, isn't the statement "the economy is in equilibrium" almost tautological? "Equilibrium" means nothing more than "whatever the model predicts". So if the model is true, then what happens must always be at the equilibrium of that model. And we use the word "disequilibrium" only when we, as theorists, are trying to understand the model and what it predicts. For example: "If X doubles, and Y didn't change, then there would be a disequilibrium in this equation/market/individual choice etc., and so the model says that Y must change too."

  4. Nick, thanks... this is an interesting point.

    As far as Tobin and the Keynes-Pigou debate is concerned, it's clear what they meant by equilibrium: a level of unemployment that persists over time because it does not induce any changes in the economy that might cause it to rise or fall. In other words, a steady state. Disequilibrium dynamics then refer to movements outside the steady state and the stability question is well defined.

    But there are dynamic equilibrium models with paths more complex than steady states, for instance Grandmont's paper on competitive equilibrium business cycles which can feature chaotic dynamics in an OLG framework. What is disequilibrium in this case? It is not defined unless one specifies some dynamics of adjustment (for instance based on learning) to describe how individuals change their plans when they turn out to be mutually inconsistent.

    In fact, I would define equilibrium in general as a path along which individual plans are mutually consistent and therefore do not require adjustment (except perhaps in response to shocks). Disequilibrium dynamics then describe how plans are adjusted when expectations fail to be realized. This is why "rational expectations" is an equilibrium hypothesis and not simply an assumption that individuals are "forward looking".

  5. Rajiv,

    Wonderful post, as always. I am pretty unfamiliar with macro but your discussion of Tobin's work reminded me of a nice paper by Tsutsui and Mino (JET, 1990). Tsutsui and Mino demontrate how a very simple linear duopoly model (with sticky prices) can generate a continuum of steady states, many of which can *only* be reached from various intervals of initial conditions.

    T-M seems to be a noteworthy contribution to the literature, and has been used extensively in various applied-theory papers in environmental economics. Thought I'd pass it along.


  6. Thanks, I'll take a look.

  7. I think this misses the point a bit. I think the solution is to say that economic models need to worry carefully about correctly identifying the dynamics whether those are equilibrium or disequilibrium. My problem with economic models is mainly that they are too static.

    By the way I think Tobin is only partly correct. Often the "shock" is internal to the economy. But because it occurs in a part of the econnomy we choose to ignore, we can't see it in our models. (i.e. Our models are missing important ingredients - over to Steve Keen and private debt levels.)

  8. Nick,
    the model is NEVER "true". It may be useful or reasonably predictive, but it is never true.

  9. I posted this in Naked Capitalism but that doesn't carry over.
    I am pleased that this analysis is finally being discussed. I am almost finished with the first book that treats far-from-equilibrium conditions in economics; Eric Beinhocker’s The Origin of Wealth, the best book on economics I have ever read. He often refers to “punctuated equilibrium” to discuss why equilibrium economics appears to occasionally work on the average but is way off base in crises.

    However, even Beinhocker fails to distinguish between assuming econometric fluctuations represent a stationary random process with meaningful future probabilities as opposed to admitting that extreme events really represent non-stationary processes which lack meaningful future probabilities. This is because the sequence of events leading up to a crisis is sufficiently complex as to be a never reapeating sequence. Thus hindcasting the past has little to do with future shocks. Well behaved equilibrium economics is somewhat like well behaved hurricane track predictions. If the boundary conditions are stable, the predictions have a zone of possible tracks which may indeed happen. If, however, the boundry conditions are not following familiar patterns, hurricane tracks like Mitch seem to wander around and are unpredictable as to their future directions.

    To differentiate between levels of equilibrium assumptions the dynamic stability of marine vessels has adopted the following definitions:

    Equilibrium, Static—situations with no unbalanced forces and no accelerations.

    Quasi-static equilibrium conditions: situations with only slightly unbalanced forces and low accelerations.

    Near-equilibrium dynamic conditions: situations where using average or periodic unbalanced forces and assuming average or periodic accelerations is the only way to analytically solve otherwise complicated problems. This includes dynamic situations which can be assumed to be periodic or a stationary random process so that Fourier Transform methods are valid in the frequency domain.

    Far-from-equilibrium dynamic conditions: significant unbalanced forces and complex accelerations make time domain analysis of the pressure and velocity field equations of fluid mechanics non computable.(This is also true for complex economics.)This includes all situations in which a non-stationary random process is present. (Note that Newton’s Second Law for rigid bodies does not have these restrictions, so body centric analysis is computable.)

    Stationary Random Process—If all marginal and joint density functions of the process do not depend upon the choice of time origin, the process is said to be stationary. A process for which the mean variances, covariance functions and probability densities are independent of time translations, i.e. Fourier transform pairs are well defined.

  10. reason - standard macro models (DSGE) today are all dynamic equilibrium models, and are meant to capture intertemporal effects. The implicit assumption is that these equilibrium paths are robustly stable under some broad class of disequilibrium adjustment processes, but the stability question is seldom formally explored. I think this is a problem because although the models are dynamic, they assume coordination on equilibrium paths by large numbers of people without considering the process by which such coordination may be attained.

    aronj - thanks for your very interesting comment, it seems that some sort of classifications such as this could be very useful for central banks and forecasters. Comments posted on NC don't transfer over (the threads are independent) but I did read your remarks when originally posted.

  11. "The very idea of a misaligned FX rate suggests that the natural state is an aligned one: perhaps the fundamentals move faster than the markets adjust, so FX is never in equilibrium."

    I don't think that it is at all unreasonable to say that it is sensible and useful to talk about the relationship between current prices in any market and how they differ from what we would expect given the fundamentals.

    It also isn't unreasonable, I think, to think that significant gaps between prices expected from fundamentals, and actual prices, to usually indicate significant and frequently highly relevant phenomena.

    The difficult, I think, flows in part from the use of the language "equilibrium" to describe prices that would be expected from simple models based upon fundamentals. This word overstates the stability of prices that are in line with fundamentals when they exist (particularly those that are nominal rather than formula based prices), overstates the gravitational pull that equilibrium prices have on actual prices, and implicitly suggests that there aren't stable prices that a far out of line with what the price one would expect from fundamentals would be.

    In markets where the fundamentals that should drive prices are well understood (for example, in the intercurrency FX arbitrage market), the markets are ruthlessly quick and efficient.

    Part of the conceptual problem, I think, flows from the tendency to see the prevailing market price as the definitionally "correct" price. While that works in a lot of practical applications, it is a poor definition for many of the purposes to which economics would like to put it, and throwing in an ill named "equilibrium price" into the mix simply further confounds the definitional mess.

  12. On first reading Schumpeter's Capitalism, Socialism and Democracy a couple of years ago I was shocked that what I saw as its quite detailed and devastating critique of equilibrium economics had been ignored by mainstream economics for over 65 years, including by Schumpeter's best students.

    Since Schumpeter's Theory of Economic Development was first published in 1911, the complacency has actually endured for almost a century.

    This is capable of generating a range of reactions, but I suspect the most common is simply to reciprocate.

  13. In Chemistry/Chem Engg, you can *effectively* shift the Equilibrium point by many approaches. At least one applies in Currency Rates 'Equilibrium':

    Keep removing one reagent and you can effectively shift the "Equilibrium" in favor (suction) of producing more of it....i.e. Keep removing (buying with XYZ Currency) dollars, and keep XYZ per USD as High as you want.

  14. Again, changing interest rates can be seen analogous to externally changing temp of the reactor, and that shifts the Equilibrium mix.

  15. Ranjiv
    I still think you missed my point. There is a difference between "Dynamic equilibrium" and Dynamic (and it is not just semantic). Dynamic means the models what agents do today given the world as they see it. One thing they cannot see is equilibrium. Dynamic eqilibrium (Note the order of the words) means we are interested in an equilibrium model but one where the equilibrium is changing. It is still not looking at the dynamics of the economy - only at the dynamics of the (implied) equilibrium (and even then not really - mostly they just extrapolate a trend). OK they add some delay in adjustment parameters and external shocks, but it still isn't a dynamic model. Equilibrium (if there is one) should be an emergent property. I'll get off my soapbox now.