dynamical systems applied to economics

Question

I'm ending my undergraduate economics course and I'd like to extend my MA research program to dynamical economic systems. Knowing that my mathematical basis is calculus of 1 and 2 variables, linear algebra, probability and statistic, differential equations and real analysis what disciplines should I take to get in in the area?

Answer 1

So for starters, I'd say to anyone who wants to do dynamical systems well (from a math perspective) you'll want to know measure theory (not sure if your real analysis was at that level or not) and topology. A little differential geometry/topology can't hurt either though some of that can be acquired on the fly (speaking from personal experience).

As for context on dynamical systems in economics (to see what it may entail), formal dynamical systems in economics is not yet really a thing, but it is a direction I think a number of people wish to take the field. Speaking as a micro theorist, much of the solution concepts from game theory are static, in the sense that while they are stable under best-response dynamics, generically you can't say much else (for example, are they attracting or repelling, etc).

A classic example is in general equilibrium theory; there isn't a real mechanism that satisfies normative conditions by which an out-of-equilibrium price vector converges to an equilibrium price vector.

A (very straightforward to a dynamical systems buff) example of some dynamics in game theory was proved originally by Milgrom and Roberts (Link). For a class of games with what amounts to complementary strategies for players, the basin of attraction of the set of equilibria under best-response dynamics is the entire strategy space. Think of this like classical Bertrand competition with homogeneous goods: if you cut prices by epsilon, I wish to just undercut you, and so on and so forth and in limit this iterated logic converges to the Nash equilibrium of zero.

Steven Smale wrote a wonderful paper discussing applications of dynamical systems theory to general equilibrium in AER (Link) which you might find interesting. He has a couple of other great papers too on game theory/economics, if you're interested I'd look them up.

Hope this was helpful and a good jumping-off point!