Why does Miles Kimball say that PhD programs rely on undergraduate Real Analysis to proxy your intelligence?

A similar question can be asked about the volunteer work that is expected in your application at universities of this caliber. Not all universities require Real Analysis, even for Masters or PhD programs. Indeed, I cannot even say with confidence that most of these programs require it.

However, a student who passes all of these courses with an A is likely ambitious, well prepared, and very intelligent. They will probably not struggle much with the math in higher economics and will have a lot of time to absorb the deeper concepts. It is a strong signal, and Mankiw can afford to be picky about who he accepts at Harvard. Kimball can be similarly selective.

If you don't take Real Analysis, there can still be a good graduate program out there for you, but admittedly, some elite universities will probably be disapproving - they are trying to discard applicants. Calibrate your sights on universities you can get into and note that you can contribute in many areas- even without real analysis. But if you have time to take the course, you should do so.

Presumably, as you have already pointed out, it's because Kimball took 9th place in a math Olympiad when he was young. It seems to have been a life-altering experience for him ...

It seems very strange to me that one would need to use Real Analysis, of all things, as a proxy for a potential to understand and make contributions to economics. I certainly don't think of mathematics as a proxy for intelligence and I think that very few people do, considering just how wide-spread the notion of a 'math nerd' actually is. Often it can just as well easily describe a very narrowly focused intelligence, which in a sense is not intelligence at all - but a kind of virtuosic performance. Richard Feynman, for example, was quite proud to say he was only above average intelligence. As one wit has said, IQ tests are very good at picking out people who can pass IQ tests. Most physicists, by the way, have very little idea about real analysis, and are quite proud to say so - and they are heavy users of math.

I mean why use a proxy? Why not simply ask students questions on political economy? Isn't this the simplest way of detecting an interest in and an aptitude for economics? Besides, many things can act as a proxy for intelligence - from art history to political philosophy apart from an interest in mathematics. But the key thing about the latter is that rarely examines it's relationship with society at large - which to my mind, is of key importance in a discipline like economics. This of course need not to be the case, and ought not to be the case, but in fact often is. Personally speaking, given how many people once they have left school say how much they hated mathematics, it would be a good proxy for educating the vast majority about technology, science and mathematics.

I recall studying some economics at high school - we had a teacher who specialised in economics - before going onto study mathematics. If the USA does not have economics tuition at high-school level, surely the intelligent thing for Kimball to do is to develop an economics curricula appropriate to college students. Presumably given his professional and institutional affiliations this would be a fairly straight-forward thing to do.

But perhaps that's too much thinking to expect from a self-confessed professional math nerd ....

Real Analysis is absolutely essential to grad school level economics. Limits are everywhere in both micro, macro and econometrics. Fixed point theorems are the basis of key foundational results in micro and macro. Indeed, both Topology and Functional Analysis (which build upon Real Analysis) appear all over all three fields.

You will not sufficiently understand this material if you have just read a few definitions and theorem statements. Actually going through exercises and proving basic real analysis results yourself is the best way to learn the material well enough to tackle grad school level texts, e.g. Mas-Colell, Whinston, Green for micro and Stokey & Lucas for macro. (Stokey & Lucas for example requires a reasonable understanding of topology. But since many mathematical intuitions carry over from real analysis, spending more time on real analysis is reasonable.)

Of course Real Analysis can be both essential, and a useful screening device to pick out students with certain skills. Proving results require creativity and strategic thinking, which are hard to otherwise measure. Additionally, the very fact that learning real analysis might be slightly unpleasant for some students provides a measure of their commitment to work hard even when it is tedious, which is essential during the PhD.