Advanced Microeconomic Theory Class Notes

Preface

These are the lecture notes for Economics 580 at Brigham Young University. This course is intended to prepare students for a first-year graduate course in microeconomics. It is based on Mas-Colell, Whinston, and Green, and it covers the core of microeconomic theory: optimization in vector spaces, the theories of demand and supply, choice under uncertainty, and general equilibrium—its examples, its welfare properties, and the existence of its equilibria.

Mathematics and Economic Theory

This course leans on mathematics more heavily than any economics course you have likely taken—not only derivatives and algebra, but vector spaces, separating hyperplanes, fixed points, and proof. I want to begin these notes with the argument for this choice. I have not chosen to use mathematics because mathematics makes economics look scientific. It is because mathematics is how economics practices science. The scientific method demands hypotheses stated precisely enough to be wrong: claims sharp enough that data—or a counterexample—could, in principle, refute them. Verbal economics slides too easily into statements that survive any evidence. A theorem cannot do this. A theorem states its assumptions in full view, and its proof shows exactly which assumption carries which conclusion. When we prove that a preference relation satisfying independence and continuity must be represented by expected utility, we have made a commitment: exhibit a decision maker who violates the conclusion and we know—precisely—which axiom of choice has failed. When the welfare theorems tell us that competitive equilibria are efficient, the proof tells us just as clearly what the result requires—convexity, price-taking, complete markets—and therefore exactly where to look when real markets fall short. This is the peculiar honesty of mathematical theory: it cannot overclaim without being caught. The mathematics in these pages is not decoration. It is the discipline that tells us what our assumptions do and do not deliver, and it is the reason the theory you learn here will still be load-bearing in every field course that follows.

However, mathematics is a language that requires time, effort, and concentration to master. It is not a natural first language for most people, and even those who seem to have mastered it can, at times, be tripped up by its complexities. For many students in this class, this will be their first sustained experience with proof—with sitting before a claim and being asked not merely to use it but to establish it. Relative to the short, usually straightforward problems assigned as homework in math classes, the arguments in these notes may seem abstract and at times baffling. Thousands of students have felt the way you may feel at the beginning of this class and have successfully mastered this material. The path to understanding things that are deep and complex has been learned over the centuries by millions. Patient, persistent effort—time spent reading, attempting proofs on your own, speaking with your classmates and the instructor, and, most crucially, repeating this process when understanding hasn’t come yet—is the path to understanding. While the marvels of YouTube and AI can deliver information more efficiently than at any point in human history, they cannot deliver understanding. That comes from the way you ingest, process, and incorporate the information you are presented.

An Invitation

Finally, a word of encouragement. The subject you are beginning is difficult, and its difficulty is not incidental—it is where the value lies. When the disciples asked the Savior why they had been unable to perform a miracle that He then performed, He answered:

“Howbeit this kind goeth not out but by prayer and fasting.” (Matthew 17:21)

Some things, in other words, do not yield to casual effort. They are of a different kind, and they respond only to a different kind of effort— sustained, deliberate, even consecrated. I have found this to be true of the material in this course. There will be proofs in these notes that do not open themselves to a single reading, problem sets that resist an evening’s work, and ideas—the separating hyperplane, the fixed point, the contingent-claims economy—that become clear only after you have wrestled with them longer than seemed reasonable. This kind goeth not out but by the intellectual equivalent of prayer and fasting: by returning to the problem again after failure, by the humility to ask for help, and by the quiet, unglamorous hours that no shortcut can replace. At this university we believe that such effort is not merely instrumental but sanctifying—that the discipline learned in mastering hard and true things carries over into every other consecrated pursuit of your life. Study this subject, then, not as a hurdle but as practice in working through hard things well. Bring your best effort, your honest skepticism, and your patience with the mathematics, and by the end of the semester you will read economic arguments with new eyes. I am glad you are here.

Scott Condie
Provo, Utah