Are business cycles chaotic? (part I)

J. Doyne Farmer

What causes business cycles?  Mainstream macroeconomics says the primary cause are “shocks”.

“If you hit a rocking horse with a stick, the movement of the horse will be very different from the stick. The hits are the cause of the movement, but the system’s own equilibrium laws condition the form of movement”

said Knut Wicksell said in 1918.  The horse is the economy, the blows from the stick are the shocks.  Absent shocks the economy would settle into an equilibrium corresponding to a fixed point in which nothing changes.  But the shocks keep coming, the horse rocks, and we have business cycles.  The model tells us nothing about the shocks, which are by definition random, but only tells us how the horse responds to the shocks.

Consider the model by Smets and Wouters, typical of those used by central banks to understand business cycles.[1] Translating the technical jargon a bit, it has seven types of shocks (changes in labor productivity, risk perception, technologies, wages, prices, spending and monetary policy).  This is typical of such models.  The number of shocks and the type of shock vary from model to model, but the basic idea is the same: shocks are inherently unpredictable.  They are outside the scope of an economic model.  People’s labor becomes more productive, people become more concerned about risk, etc, but studying these root causes is off the table.  All the models do is explain how the world responds to these changes, changes which we do not attempt to describe.

Shocks are thus merely a reflection of our ignorance.  They are the things that influence the world that we don’t understand.  Because we don’t understand them, they appear as if by magic so prediction is ruled out from the outset.  All we can predict is the aftereffect – how the system’s equilibrium laws condition its movement after the stick hits it.

To someone like me, with a dynamical systems background, this is a very strange state of affairs.  To make an analogy, consider the weather.  Here in England it can be sunny now, then rain, then be sunny again, then rain again.  Is this cycle caused by shocks?

No!

It is caused by the fact that the equations of fluid flow are highly nonlinear.  The forcing of the sun drives the system away from equilibrium, which engages the nonlinear behavior of the system and causes it to oscillate in an irregular manner.  It does this all by itself, without any need for external shocks.  This is called chaos.  A chaotic system responds to shocks, and can even amplify them enormously, but it doesn’t need them to generate random-looking dynamics.  Even when left on its own, things never settle down.

“A chaotic system responds to shocks, and can even amplify them enormously”

The economy is also highly nonlinear.  Can economists use this to explain business cycles?  In fact, back in the 90’s there was an interest in the possibility that chaos might be the cause of variability in economic patterns.  People formulated equilibrium models with chaotic dynamics.  But economists looked for chaos in the data, didn’t find it, and the subject was dropped.  For a good review of what happened see Roger Farmer’s blog[2].  Except for a few brave souls[3], chaos has been virtually ignored in economics ever since.

This is an unfortunate story.

The tests that were run were not really tests for chaos, but for “simple chaos”, and the wrong lesson was learned.  Simple chaos is low-dimensional chaos that only has a few degrees of freedom, so that that the state of the system at any given time can be described by just a few numbers.  If we apply the same tests to the weather, they also fail to detect simple chaos – the endogenous oscillations of the weather are much more complicated than simple chaos.  For lots of other reasons, both experimental and theoretical, we know that the weather is a chaotic system.  Like the weather, the economy is complicated.  To make matters even worse the economy is nonstationary – it evolves through time as institutions and technologies change become richer and more complex.  So it is not surprising that simple chaos was not found in the data.  That does not mean that the economy is not chaotic.  It is very likely that it is and that chaos can explain the patterns we see.

Most of the  Dynamic Stochastic General Equilibrium (DSGE) models that are standard in macroeconomics rule out chaos from the outset.  Production models that are used by banks, such as the Smets-Wouters model, are forced to be linear to make their parameters easier to estimate.  Linear models cannot display chaos – their only possible attractor is a fixed point.  The rocking horse framework is enforced by definition.  Such an economy can never generate a business cycle on its own – it can only respond to unpredictable stimuli supplied from the outside.  Remarkably a standard family of models is called “Real business cycle models”, a clear example of Orwellian newspeak.

“In a future blog post I will argue that an important part of the problem is the assumption of equilibrium itself”

In a future blog post I will argue that an important part of the problem is the assumption of equilibrium itself.  While it is possible for an economic equilibrium to be chaotic, I conjecture that the conditions that define economic equilibrium – that outcomes match expectations – tend to suppress chaos.  Ironically, if business cycles are chaotic, we have a chance to predict them.   At least we haven’t taken the possibility off the table.

There is a broader lesson here.  Results that appear to close off an otherwise interesting line of attack and argue that something is impossible should always be examined very carefully.  The fine print can be

[1] https://ssrn.com/abstract=1687574

[2] http://www.rogerfarmer.com/rogerfarmerblog/2016/10/4/nho932exasra0c2a2amkvdmovcy9rz?rq=chaos

[3] http://www.sciencedirect.com/science/article/pii/S0165188998000116