Math major looking for some real-world connection

[ricardo]

Hi, i am a math major from Chile, i really love math, but i want to use advanced math in something more connected with real world that pure math, my school is very purist in his teaching style.

 

I wonder which way mathematics beyond real analysis are applicable to economics, i say things like Measure Theory, Probability Theory, Abstract Algebra, Algebraic Geometry, PDE, Complex and Functional Analysis, Topology and Dynamical Systems are applied to economics.

 

Are advanced math used in econ research?

Are this things part of the day to day tools of a economist?

[Python7768]

Yes, in my opinion, the people who are doing the most interesting research in economics are currently developing mathematically rigorous models and then joining that with some kind of data. I think many if not all of those topics are used within economics (some more than others). Probability theory shows up literally all over the place and is a viable contender for having been the most useful undergraduate class I took in preparation for my first year.

 

One example that could use several of the things is a simple macroeconomic model. You will typically have a measure of heterogeneous agents who are maximizing expected utility of their lifetime. In order to solve the majority of the models that look like this, you will need to use at least a little of each of the following: Measure theory(mostly in the context of probability), Probability theory, Some kind of difference or differential equation, Functional analysis (at least to prove the existence of a value function), etc.

 

If you aren't afraid of the math; Stokey, Lucas, with Prescott's Recursive Methods in Economic Dynamics is an excellent book. It is 20 years old, but I think it gives a good introduction of the use of mathematics in macroeconomics. There are lots of other applications in micro and game theory. If you want to see how it is currently being used then you can probably crack open a volume of Econometrica or the American Economic Review. Those should have some good examples.

[tm_member]

Theorists use these tools all the time but it is mostly abstract work on solving for difficult equilibria in multiple dimensions. Topology, complex analysis, etc. has virtually no role in applied economics. Some people find the mathematically rigorous models to be valid and try to fit the data to the model in order to discover structural relationships between variables. Others feel that the data doesn't meet the assumptions of these models and is a waste of time and effort.

 

Applied economists generally use simplified models to derive hypotheses they can test but rarely does there exist a real world analog that is close enough to the assumptions underpinning the abstract work for it to have an application.

[yankeefan]

Ricardo, I believe you're going about this the wrong way. Although we often joke about economics being a subfield of applied math, one should not pursue a PhD in economics if they are only interested in using math to solve real world problems. Math is (ab)used in economics to express economic theory in a formalized manner in order to give it coherence and [internal] consistency, as well as to deduce new hypotheses. And as tm_member pointed out, most of the theorizing that takes place uses assumptions that are rarely satisfied in the real world. If you are interested in [directly] mathematically modelling real world problems and systems, you should consider an applied mathematics PhD. If you have strong programming skills and an affinity for scientific computing, you may also want to consider a PhD in operations research or a hybrid computer science/business PhD (something along the lines of NYU IOMS).