In school I learned the basics of reading, writing and arithmetic reasonably well. That may be partly due to competent teachers, a stable family and school environment, and my being somewhat diligent. However, I am convinced that I would have learned a whole lot more if I had had access to the enormous number of excellent teachers and the virtually infinite amount of content on every conceivable subject we have available today: not in person but over the internet.

Though I am not very good at it, I like mathematics a lot. Over the years, I was required to learn some bits. In my undergraduate engineering classes, I learned the calculus and some linear algebra but nothing to write home about. Then while studying computer science, I learned an entirely different area of mathematics: discrete maths, particularly combinatorics. Then for my post-graduate work in economics, I got to learn a lot more of the calculus, and some statistics (because of econometrics, a subject that I hate with uncharacteristic passion) and probability theory.

In the first grad-level course on price theory (aka microeconomics) I took (Econ 201A) and I learned quite a bit of mathematics. We were following the graduate level textbook by Mas-Colell, Whinston and Green for the class. Take my word for it — you are lucky if you don’t have to open it. To me it was fairly impenetrable at first but I got a hang of it eventually. In one class, the professor wrote out the proof of “the second fundamental theorem of welfare economics.” It states that under certain restrictive conditions, any Pareto efficient allocation can be achieved as a competitive equilibrium if you do appropriate lump-sum transfers and let the market grind out the resulting equilibrium. The first fundamental theorem of welfare economics states that every equilibrium that undistorted markets attain is Pareto efficient.

In other words, the first theorem says that markets work, and the second theorem says that whatever you want to achieve, you can achieve through the market if you are clever enough to figure out the initial conditions (endowments) necessary. Alternatively stated:

- The First Welfare Theorem: Every Walrasian equilibrium allocation is Pareto efficient.
- The Second Welfare Theorem: Every Pareto efficient allocation can be supported as a Walrasian equilibrium.

Of course, I have forgotten the proof of the 2nd FToWE.[1] What I remember for certain is that it involved Kakutani’s fixed point theorem, and somewhere along the way I learned Brouwer’s fixed point theorem too. Fixed point theorems relate to topology, a subject concerned with “properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.” I figured I understood what fixed point theorems were. Or maybe not.

I should really get to the point that I am trying to make. Fixed point theorems are exciting and fun. They make you go “Aha!” with amazement. Here’s a video that does an excellent job of introducing the idea of fixed points. (I suggest you skip to 2:00 time stamp in the video below.)

Alright, I assume that you are going to skip watching that video for now, and I hope you get to it sometime. I confess I am fixated on fixed points. The image at the top of this piece is an illustration of Brouwer’s Fixed Point theorm from a nice piece titled Combed Coconuts have Cowlicks.

Fact: There is at least one point of the crumpled sheet that remains precisely atop the corresponding point on the lower sheet. In a sense, regardless of the crumpling, this point hasn’t moved. Said in mathematical terms, this is Brouwer’s Fixed Point Theorem: a continuous mapping of a compact convex set onto itself has a fixed point.

If you are curious about fixed point theorems, you find all sorts of material on the web — text, graphics, video — that explain it to whatever depth and detail you desire. Not just about fixed points, all sorts of mathematical subjects.[2] Heck, why just mathematics — you will find content on whatever subject you fancy.

Here’s the point of this part of this series: education has changed. The way we learn and the way we teach has changed. The who learns has changed, and who teaches has also changed. Education has changed not just from the way it was done 200 years ago, but even how it was done 20 years ago. I have been learning not in class rooms, not from hardcopy books, under someone else’s direction on what I should learn — but in the comfort of my own home, learning whatever I want to learn, subjects that I find interesting, taught by people who are not “accredited” by some bureaucracy but by their “subscribers.” When I see that the Vsause channel has 16+ million subscribers and that each video is has been watched millions of times, that’s sufficient accreditation for me.

There are no barriers to entry in education and learning any more. Nobody is going to ask you to show proof that you have the prerequisites to learn something.[3] Nobody is going to ask you to show proof that you are certified to be a teacher on the web. Just bring your curiosity and your thrist for knowledge and understanding, and you will learn.

Countries that refuse to change with changed circumstances are likely to suffer. I am afraid for India and other countries that have uneducated leaders, who are stuck in some ancient time, who are only interested in extracting rents from their control over the education system. Their control condemns hundreds of millions to abject poverty. They must be eliminated.

In a later part, I will outline how a good education system can evolve to fully respond to the altered circumstance.

*{Previously: Parts One, Two, Three, Four, Five, Six.} *

### NOTES:

[1] For a sketch of the proofs, see this.

[2] Here’s one that I watched this morning. It’s from Mathologer and the subject is The Hardest “What comes next?”

[3] If I had gone to an Indian university and told them that I’d like to enroll for a PhD in economics, they would have laughed at my face because I didn’t have a single course in economics, never mind an undergrad or postgrad degree in economics. Fortunately, in the top schools in the US, they care more about how motivated a student is, and not so much about certificates.

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