How Much Math Do Economists Really Know?

Murray Rothbard once wrote that mathematicians don't take seriously the math used by economists. It is pretty basic unimpressive stuff. Most would have trouble getting even as far as multivariable calculus.

During a commencement speech delivered at Chapel Hill - Chauncy Hall School by Greg Mankiw, the chairman of the economics department at Harvard University, Mnakiw seemed to support Rothbard's view (Rothbard, btw, received a bachelors degree in mathematics from Columbia)

Mankiw said:

Fast forward to my own high school graduation[...]At the time, I was the school math geek. I took all the hardest math classes, took more math classes on the weekends at a nearby university, spent the summer before my senior year at a summer activity focused around math and astrophysics, and won the school math prize. I thought I was pretty hot …..Well, you get the idea.

When I went to college the next fall, I started off as a math major, thinking I would end up being a professional mathematician. I was doing what economists call pursuing your comparative advantage, which means doing what you are good at compared with other people. I thought if I was so good at math compared with my high school classmates, it would make sense to turn that talent into a career.

But then something happened: I met some other students who were really good in math. And I mean really good. These were the kind of kids who not only took hard math courses in high school and did well in them, but they spent their free time competing in the international math Olympiad. They were in an entirely different league than I was. I felt like I was the most valuable player on my little league team, and all of a sudden I was practicing with the Red Sox.

Over time, I realized that I was pretty good in math, but far from a star. I was good enough to take college-level math classes and pursue a more quantitative career, but I was probably not cut out to become a professional mathematician[...]

During my freshman year, I started dating a young woman, who happened to be on the same dorm floor as Richard and I. She also happened to be taking a freshman course in introductory economics. Those coincidences changed my life.

She used to come back from her economics class and tell me what she was learning. To my surprise, I found it fascinating. I entered college with little idea what economics was, and little intention to study it. But from what she told me, it seemed that what she was learning was far more interesting than anything I was learning in any of my classes.

So the next semester I started taking economics classes. And I really liked them. And, it turned out that I was pretty good at it.

Eventually, I switched my major from math to economics. I went on to get my PhD and have been a professor of economics at Harvard for almost 30 years, as well as an economic adviser to presidents and presidential candidates.

Here's Rothbard's conclusion about the math used by economists, like Mankiw, who use a lot of mathematical equations (though are generally severely limited in math understanding)

Actually, mathematical methods necessarily introduce many errors and inanities[...]The best readers' guide to the jungle of mathematical economics is to ignore the fancy welter of equations and look for the assumptions underneath. Invariably they are few in number, simple, and wrong. They are wrong precisely because mathematical economists are positivists, who do not know that economics rests on true axioms.

The mathematical economists are therefore framing assumptions which are admittedly false or partly false, but which they hope can serve as useful approximations, as they would in physics. The important thing is not to be intimidated by the mathematical trappings