Week 4 in Review

I wish I didn’t skip week 3 review. Now I don’t remember what I learned that week.

1. Economic Analysis

The problem set was a killer. I had to solve models, find data, graph data, and read articles. It was vey time consuming, but on the other hand it was somewhat enjoyable. The lectures also have become little more interesting as models are less arithmetic and more controversial. We covered social security and public debt, discussed Ricardian equivalence and fiscal multiplier.

2. Game Theory

We finished discussing Bernoulli function and began to talk about mixed strategy equilibrium. I am stating to think that game theory notations are ugly. The lectures have been more or less standard, but I am looking forward to the proof of existence of a Nash equilibrium for any game.

3. Statistics

Finished the chapter on joint distribution and began the one on expected value. Joint distribution was somewhat difficult; I feel I need to review my multivariable calculus. I was introduced to St. Petersburg paradox and realized that gambling contributed quite a lot to human knowledge.

4. Real Analysis

The midterm was the most difficult exam I have done so far. There were three parts to the exam, and I didn’t even get to read the third part. Anyways, finished the chapter on integration.

Week 2 in Review

1. Economic Analysis

I am increasingly inclined to think that macroeconomics isn’t as fun as microeconomics. What’s the big deal about Laffer curve, anyway? (I am a fan of Martin Gardner.) I do enjoy the algebra, though. Solving the first order conditions and deriving the steady-state consumption level is like doing a Sudoku puzzle; mind-numbingly fun.

Anyways, the point of this post was to review what I learned, like “lump-sun taxes are non-distortionary and consumption crowding-out” or “proportional-tax model predicts that the revenue maximizing tax rate is equal to the labor share in production which is empirically around 2/3”. Yawn.

2. Game Theory

The challenge problem of this week was to derive a condition analogous to WARP on a choice function such that a choice function generated by a quasi preference is satisfies the condition and a choice function satisfying the condition is quasi preference generated. A quasi rational preference is transitive and irreflexive, such as “xPy if x-1>y” where x,y are integers. Of course, I have not solved the problem (yet).

We also covered Gibbard-Satterthwaite theorem and started to build the formal concept of game theory.

3. Statistics

We departed with probability and entered the world of random variable and discussed several important discrete random variables, like the Bernoulli, binomial, geometric and Poisson. I should review these.

4. Real Analysis

The “fun” problem of the first assignment had nothing to do with real analysis, but was still interesting: A set of points in a plane and a subset of all the line segments connecting the points are given. Each point represents a switch. If a switch is flipped, then all other switches connected to that switch (via the line segments) are flipped. Initially all the switches are turned off. Prove you can turn on all the switches.

Apparently the problem can be brought into the realm of linear algebra and solved, but it seems there is a different approach as well. I was thinking about using Euler characteristic on planar graphs. I am yet to see if this goes anywhere.

We covered the definition of Riemann integral and the intuition of Lebesgue integral.

Week 1 in Review

I had an ambitious goal of keeping a daily journal of what I learned, but my ambition died the first day and I decided to keep a brief weekly journal instead.

1. Economic Analysis

First lecture was a review. The main point of interest was that the amounts of labor and consumption are equal in a single-sector economy (Robinson Crusoe economy) and in a two-sector economy (decentralized market economy) given the same preference and technology. The natural question is, of course, why that is true. I frankly couldn’t answer the question and embarrassingly I still have not found a satisfactory answer. A point I need to think a little more.

The second lecture was the beginning of a tax analysis, and since we did not finish the model’s analysis, I have an excuse to delay a log on it.

2. Game Theory

The lectures were very satisfying in that they provided mathematically vigorous definitions of preference relation and choice function. Ever since I took the real analysis class, I wanted some exposure to mathematical rigor in economics, and this class seems to provide just that at the right level. The core of the first week’s lectures was proof of Houthakker’s theorem, which is interesting enough that I may write a post about it later.

3. Statistics

So far, we only had somewhat boring lectures on probability. There was one interesting question given during the class, which was pretty elementary but still quite confusing: There are 49 balls in a jar, each numbered from 1 to 49. You choose 7 numbers and 6 balls are drawn. What is the probability that exactly 3 balls drawn show the numbers you chose? There are 49C6 ways in which 6 balls are drawn from the jar. Of those 6, we want 3 balls to have numbers you chose; there are 7C3 ways that 3 drawn balls match with the 7 chosen numbers. The remaining 3 balls should have numbers you have not chosen and there are 42C3 ways in which 3 balls do not match the numbers. So the probability is (7C3)(42C3)/(49C6), which is around 0.0287. The curious part of the problem is that one can think in terms of numbers you choose instead of balls you draw; this yields the probability of (6C3)(43C4)/(49C7) which is seemingly different from the previous one, but is in fact the same.

4. Real Analysis

I thought the second quarter would be better and I was wrong. This is the class that I feel very lost during the lecture so I should really review the materials on a daily basis, but I have not even done that. So far, we have proved that differentiable implies continuous, chain rule, mean value theorem, and l’Hopital’s rule. Sadly, I don’t think I fully understood any of them. The problem set is also discouraging and I have not touched it yet, so not much to discuss yet.