"Curiously, though economists like to intimidate other social scientists with the mathematical rigor of their discipline, most economists do not have this level of mathematical education...

...One example of this is the way economists have reacted to 'chaos theory'... Most economists think that chaos theory has had little or no impact - which is generally true in economics, but not at all true in most other sciences. This is partially because, to understand chaos theory, you have to understand an area of mathematics known as 'ordinary differential equations.' Yet this topic is taught in very few courses on mathematical economics - and where it is taught, it is not covered in sufficient depth. Students may learn some of the basic techniques for handling what are known as 'second-order linear differential equations,' but chaos and complexity begin to manifest themselves only in 'third order nonlinear differential equations.' Steve Keen (2011).

Debunking Economics: The Naked Emperor Dethroned?, Revised and expanded ed. Zed Books, p. 31

The above quotes are also in the first edition. Before commenting on this passage, I want to re-iterate my previously expressed belief that some economists, including some mainstream economists, understand differential and difference equations.

I misremembered this comment as being overstated for polemical purposes. But, in context, I think it is clear to those who know the mathematics.

I took an introductory course in differential equations decades ago. Our textbook was by Boyce and DiPrima.
As I recall, we were taught fairly cookbook techniques to solve linear differential equations. These could be first order or second order and homogeneous or non-homogeneous. They could also be systems of linear differential equation. I recall some statement of an existence theorem for Initial Value Problems (IVPs), although I think I saw a more thorough proof of some such theorem in an introductory^{1} real analysis course. We might have also seen some results about the stability of limit points for dynamical systems. Keen is **not** claiming that economists do not learn this stuff; this kind of course is only a foundation for what he is talking about.

I also took a later applied mathematics course, building on this work. In this course, we were taught how to linearize differential equations. We definitely were taught stability conditions here. If I recall correctly, the most straightforward approach looked only at sufficient, not necessary conditions. We also learned perturbation theory, which can be used to develop higher order approximations to nonlinear equations around the solutions to the linearized equations. One conclusion that I recall is that the period of an unforced pendulum depends on the initial angle, despite what is taught in introductory physics classes^{2}. I do not recall much about boundary layer separations, but maybe that was taught only in the context of Partial Differential Equations (PDEs), not Ordinary Differential Equations (ODEs). This is still not the mathematics that Keen is claiming that economists mostly do not learn, although it is getting there.

You might also see ordinary differential equations in a numerical analysis course. Here you could learn about, say, the Runge-Kutta method. And the methods here can apply to IVPs for systems of non-linear equations^{3}. I believe in the course that I took, we had a project that began to get at the rudiments of complex systems. I think we had to calculate the period of a non-linear predator-prey system. I believe we might have been tasked with constructing a Poincaré return map.

According to Keen, a sufficiently numerate economist should know the theory behind complex dynamical systems, chaos, bifurcation analysis, and catastrophe theory^{4}. I think such theory requires an analysis able to examine global properties, not just local stability results. And one should be interested in the topological properties of a flow, not just the solution to a (small number of) IVPs. Although this mathematics has been known, for decades, to have applications in economics, most economics do not learn it. Or, at least, this is Keen's claim.

Economists should know something beyond mathematics. For example, they should have some knowledge of the sort of history developed by, say, Fernand Braudel or Eric Hobsbawm. And they should have some understanding of contemporary institutions. How can they learn all of this necessary background and the needed mathematics^{5}, as well? I do not have an answer, although I can think of three suggestions. First, much of what economists currently teach might be drastically streamlined. Second, one might not expect all economists to learn everything; a pluralist approach might recognize the need for a division of labor within economics. Third, perhaps the culture of economics should be such that economists are not expected to do great work until later in their lifetimes. I vaguely understand history is like this, while mathematics is stereotypically the opposite.

**Footnotes**

- As a student, I was somewhat puzzled by why my textbooks were always only
*Introductions to X*or*Elements of X*. It took me quite some time to learn the prerequisites. How could this only be an introduction? Only later work makes this plain. - Good physics textbook are clear about linear approximations to the sine function for small angles. Although our textbook motivated perturbation theory in the context of models of solar systems, I have never seen perturbation theory applied here in a formal course. Doubtless, astrophysicists are taught this.
*Stiff differential equations*is a jargon term that I recall being used. I do not think I ever understood what it meant, but I am clear that the techniques I think I have mostly forgotten were not universally applicable without some care.- Those who have been reading my blog for a while might have noticed I usually present results for the analysis of non-linear (discrete-time) difference equations, not (continuous-time) differential equations.
- There are popular sciences books about complex systems.

## 9 comments:

You wrote: "How can they learn all of this necessary background and the needed mathematics[5], as well?" But footnote [5] is missing...

Having taken some (basic) applied maths, I was shocked at the way differential equations were 'used' in my mathematical economics classes. There's a strange focus on equilibrium properties over actual exploration of the dynamics of the system.

"Students may learn some of the basic techniques for handling what are known as 'second-order linear differential equations,' but chaos and complexity begin to manifest themselves only in 'third order nonlinear differential equations.'"

Uh... say what?

In my browser, footnote 5 is there. Maybe I mucked up the HTML.

I suppose some of the dynamic concepts can be introduced at an introductory level.

That bit about "second-order..." is apparently known as the PoincarĂ©-Bendixon Theorem.

Yes, but that's about dimension not order. Most well known chaotic systems are first order (for example, Lorenz). And in a way that's the whole point of "chaos" - that really complex dynamics can result from fairly simple relationships.

(Also, with discrete time difference equations you don't even need three dimensions, and that's pretty much what most models use, not just in economics)

I've demonstrated awareness in past posts of the possibility of chaos in discrete time systems.

Anyways, a clear connection exists between the order of a single ODE and the dimensions of a system of differential equations.

Let

f(x''',x'',x',x) = 0 be a third-order ODE. Then:y=x'z=y'f(z',y',x',x) = 0is a system of three first-order differential equations.

A forced, damped pendulum, apparently, is a counter-example to Keen's claim. It is anon-homogeneous example, if I understand correctly; and it is a system defining a flow in a three-dimensional space, anyways.

3Yes, there is a connection but it's not the same thing. You can reduce a nth order DE to a system of n 1st order DEs. Is there an "only if" arrow going the other way (I actually don't know, but I suspect, no)?

And that sort of highlights that you don't need to study 3rd+ order DEs to play with chaotic systems. Really all you need is to know that higher order DEs can be reduced to 1st order systems. Which is why standard ODE courses focus on 1st and 2nd order DEs.

I'm not disagreeing with you, just Keen.

BTW, I saw a paper not too long ago where something like what you outline in your "The Moving Finger Writes..." post was implemented in a laboratory/classroom setting. The motivation for the paper was that there was a disagreement between the co-authors, one of whom believed that agents would find *some* way to get to the interior, unstable, equilibrium. The other one thought it was going to go wacky, as the tatonnment dynamics suggest. In the experiment, it went wacky with price dynamics unraveling away from the equilibrium. I wish I could find it but it was just something I noticed in passing.

Using Google, I find Sean Crockett's Price Dynamics in General Equilibrium Experiments surveys experiments, including with the Scarf instability example.

It wasn't that, it was a working paper, and I can't find it now for the life of me.

I brought it up because in the discussion on "The Moving Finger Writes..." I speculated that agents would find some way to stay at the unstable equilibrium. This working paper seems to have shown that is not the case.

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