Mathematical Economics

Krugman has criticized his fellow macro economists for falling too much for mathematical beauty in theory, but exactly what is mathematical economics, and is any interesting mathematics coming out of it? Well game theory has certainly found some of its most important applications in economics, but it didn't exactly originate there. So what the heck do economists do, mathematically?

A lot of the same things physicists study, for a start: calculus, linear algebra, differential equations. Mostly though, at least at the elementary level, they like optimization. Optimization brings questions of convexity, so there is at least a hint of geometry.

Dynamical systems theory seems like a natural match, so I suppose that at least some economists do that, though the mathematical econ book I have doesn't seem to mention the subject. I have heard claims of links to gauge theory, and would be curious to know how that works. Surely symmetry and group theory must be relevant somewhere.

The most important point of a mathematical theory is that it ought to give concrete predictions. How is econ doing in that regard?

I would be interested if anyone has pointers to some of the main points and issues of mathematical econ.

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