It has been an interesting week. The microeconomic sequence that all Ph.D. students must go through may seem analytical, abstract, or plain torturous to people. To me, I am getting a sense of awe. Let's just give an example

The 3 implications of utility maximizing behavior are:

1) The demand functions are homogenous to degree zero

2) The Slutsky substitution terms are symmetric

3) The Slutsky matrix of substitution terms is negative semidefinite.

All these concepts may seem alien but they are "painfully" derived from hard core mathematics. As I was going through some of the proofs, I suddenly was reminded of this saying from Isaac Newton, which I first read in Paul Samuelson "Foundations of Economic Analysis":

What Newton is saying here is that science, unlike art, is cumulative. Actually, I see this microeconomic sequence as more than just pure analytical economics, to me, it is also a course in the history of economic thought!

"Now wait a minute, what do u mean? It is just pure torturous mathematics to me," you might say. Well, the earlier giants of economics, John Stuart Mill, William Stanley Jevons, Alfred Marshall, John Hicks, Pareto, Debreu, Koopmans, Kenneth Arrow and Paul Samuelson, had went slowly but surely build economic theory into this marvelous piece of gem that is now microeconomics. We used to think that utility is cardinal, that it should be "obvious" that demand curves slope downwards etc. It led to decades of arguments and confusion before, finally, consumer theory is supposed to be settled, with the grand theory of "revealed preference".

Now, a person by the name of Kelvin Tan is now studying consumer theory. I can be said to be standing on the shoulders of these giants. I just can't help but feel awed as I laboriously prove the various theorems of duality, quasiconcavity, convexity etc. Going deeper into microeconomics, such as how the separating hyperplane theorem illustrates vividly how Adam Smith's classic free market idea works in decentralizing producers and consumers decision fills me with such a sense of deep respect for these "giants" of economics.  

To give a more practical example how I am benefiting now as I stand on the shoulders of these giants, let's just discuss the Envelope Theorem. Just imagine that, lemmas such as Roy's Identity, Shepard's Lemma, and Hotelling's Lemma can all be proved so simply using Envelope Theorem. I cannot imagine the distant past when Jacob Viner and his draughtsman argued endlessly over the curvature and the tangency between short and long run cost curves, which was finally solved once and for all by Paul Samuelson. Certainly, solving problem sets involving these lemmas have never been easier once you understand how to use Envelope Theorem.

Of course, my honors year in NUS introduced me to Envelope Theorem but, due to my limited exposure to economics then, it was a very confusing experience. I remember many a times, Xing Xiaolin, my lecturer in Advanced Economic Analysis would remark to us in lecture on how to prove all the above lemmas as, "Just use Envelope Theorem," leaving all of us blur like sotong. At that time, the lecturers give such a brief description of the Envelope Theorem that is hardly enough for us to prove all those lemmas.

Finally as I undertake my PhD course, I understand how useful Envelope Theorem is and how to finally apply it. I do remember one of my honors classmate, Brian Ang, remarking that, once he finished this honors course, he "die die" must go and take an overseas Masters, so as to make all the suffering we are going now in honors worth it. This is because we realized then that the NUS economic honors course is equivalent to the US Master's course in coverage.

Indeed, reflecting on my standing on the shoulder of earlier giants of economics, I sometimes feel like how Moses felt when the Lord appeared to him in a burning bush; that I am now standing on holy ground.  The feeling is almost like having Adam Smith and Alfred Marshall whispering to me, "Psss, take off your sandals, for you are standing on Holy Ground."  Or like how Chinese gongfu masters like Zhang Wuji felt in reading manuals like "Jiuyang Shenggong" :-)  

Haha, time to get back to reality. It is such a pity that most of the students here in my department does not feel the same way. In fact, what made me kind of amused is that most of the Chinese Nationals in my department are now switching fields to Statistics, rumors being that they originally came here in the US primarily to get a job, which a Masters in Statistics allow them to do so easily, thus, many of them have no intention at all to finish their Ph.D. A non-Chinese made this statement to me. Sigh, this I guess is another reason why we Chinese have such a bad reputation overseas.

But never mind, God created me to be unique and idealistic. I am aware that just being in awed by the sacredness of consumer theory will hardly help in publishing anything in economic journals. All I can do is to cling on to the promise from Paul Samuelson, the "high priest" of economic science, that consumer theory is the rock in which every other branch of economics is built, such as international economics, public finance, etc. Thus, once I fully mastered the generality of consumer theory, I would merely need to apply the various special cases into these other fields. At least, that is the promise I seem to get from reading Samuelson's "Foundations of Economic Analysis".

They say a scientist needs to be passionate about his subject to be a true scientist. Well, I will see how far God wants to lead me in that. :-)

For economic students keen to understand more about how mathematics apply to economics, the first book I would recommend is T J Koopmans, Three Essays on the State of Economic Sciences. That book gives a very readable literary introduction to how linear algebra, which for example concepts like compactness, bounded and closed sets, are used in basic economics. It also gives a very readable introduction to the separating hyperplane theorem that I have described earlier.

Once you have had enough of that book, the second book would be the classic Foundations of Economic Analysis by Paul Anthony Samuelson, most people would agree that he is the guru of economic science. That book contains a higher degree of mathematics but it is still fairly readable for someone with a 4-year degree in Economics. Robert E Lucas started off his economic studies with this book J .

At this moment, I can see the connection between the above 2 books. Koopmans's book uses linear algebra in economics while Samuelson uses calculus in his. In the opening pages of Three Essays, Koopmans explicitly linked his book as a further building up on Samuelson's Foundations. So, enjoy yourselves J .