# Microeconomic Theory I: General Equilibrium and Cooperative Games

## Contact

Instructor: Marcin Pęski, mpeski@gmail.com. Office hours: Tuesday, 2.30pm-4.30pm, GE207.
TA: Ruizhi Zhu. Tutorial Th, 2-4pm, FE326. There will be 5 tutorials. The first tutorial will take place on October 31 and the last on Dec. 5.
Lecture: Tue, 9-11pm, AB114, Th:, 9-11pm, BF323. We are going to start the lecture at 9.00am sharp (not 9.10) .

The required textbook is Microeconomic Theory by MasCollel, Whinston and Green (MWG for short). Below, you can find a more detailed description of the topics together with required readings. The assignment of topics to particular days is tentative and it may move as we go along. Another great book is “Microeconomic Foundations I” by D. Kreps.
 Date Topics Lecture notes Textbook Problem sets Due date Solutions Feasible allocations. Mathematical notation. Production economy. Feasible allocations. Examples. Randomizations. Production economy MWG 15.A-C,16.A-B Problem Set 1 31.10 Solutions to PS 1 Pareto-optimality. Definition. Utility-possibility frontier. Examples. Assignment problem. Quasi-linear preferences. Pareto-optimality MWG 16.E-F Walrasian equilibrium. Ownership structure. Prices and demands. Definition of equilibrium. Examples. Equilibrium MWG 16.B, 17.A-B Problem Set 2 31.10 Solutions to PS 2 Existence of equilibrium. Fixed-point theorems. Existence Theorem and proof. Equilibrium MWG 17.C Welfare Theorems. First Welfare Theorem. Separating hyperplane theorems. Walrasian Equilibrium with transfers. Second Welfare Theorem. Core. Welfare Theorems MWG 16 C,D Problem Set 3 14.11 Solutions to PS 3 Multiple markets. Radner equilibrium. Examples. Equivalence between Walrasian equilibrium and Radner in the model with money. Uncertainty Problem Set 4 21.11 Solutions to PS 4 Trading under uncertainty. Arrow-Debreu model of uncertainty. Radner with Arrow-Debreu securities. Incomplete markets. Uncertainty MWG 19.A-F Introduction to cooperative theory. NTU and TU cooperative games. Solution. Core. Nash bargaining. Shapley value. Cooperative Games MWG 18. Appendix A Problem Set 5 28.11 Solutions to PS 5 Matching I. Marriage matching problem. Stable matching. Gale-Shapley algorithm. Efficiency. Rural hospital theorem. Strategy-proofness. Matching Matching II. Matching with transferable utility. Other matching problems. Problem Set 6 03.12 Solutions to PS 6 Final. List of required results 13.12, 2-4pm. Room: WW 126

## Objectives

This is the second of four parts of the Ph.D microeconomics sequence. The class has two objectives.
1. The primary objective is to introduce you to the some of the foundations of the microeconomic theory. The class is devoted to studying economic allocations. We spent at least half of the course talking about allocations in the Walrasian economy, including their feasibility, efficiency, equilibrium, etc. The rest of the class will be devoted to allocations in other problems, including matching. If we have time, we will talk about comparative statics. Most of the material will be new to you (unless you have taken the graduate course before).
2. The second, and more universal, objective of this course is to introduce you to the formal approach to the economic argument. The class is proof-based and most of the lecture is going to proceed in the rhythm of definition-theorem-proof-example. You will learn how to read the proofs and how to carefully write them.
Because the topics are either new, or approached in a novel way, and because most of you haven’t seen a proof-based class before, many of you will find that it is a difficult class, perhaps the most difficult course you have ever taken. It is essential that you allocate a sufficient amount of time to study for this course, and that you study in a right way. You can find some advice how to do it below.

## Prerequisites

I assume that you have all taken or otherwise are familiar with the material covered in the math refresher course ECO1011, L0101 Mathematics and Statistics for PhD and MA Doctoral Stream Students. And, of course the first part of Micreconomics I.

The grade for my part of the class will consist  ≥ 90% from the exam grade and  ≤ 10% from the participation in the tutorial.
The exam will consist of four equally weighted questions. I will write the questions together with 4 questions that I am supposed to prepare for summer comps. Then, I will (more or less) randomly choose the questions for the midterm and the questions for the comps. In this way, you will know that the midterm and comp questions are similar with respect to their type and their difficulty.
I post past midterms on the course website. You can also find the past comps (but I don’t remember where - either through the library or some departmental site). Typically, there are few types of questions:
• reproduce a definition, a statement of a result, or a proof that you learned in a class,
• prove some consequence of a definition, or derive some property of a given model, or verify that a given definition is satisfied in the model.
• Solve something: describe the Pareto set, find an equilibrium, etc.
In the tutorial, I like the idea that the solutions to the homework are presented by students. The TA should step in only if nobody solved the problem correctly (which is not going to happen very often), or if he wants to present an alternative method. In the beginning of each tutorial, (or perhaps, in the morning of the tutorial day), a TA will ask you to report which questions you have managed to solve. Then, the TA will choose a volunteer to present a solution from among the people who reported the solution. Your grade from the tutorial will depend on how many problems you reported and how well you managed to present the problem to the class. The exact rules will be determined by the TA.

## How to study for this class

I want to emphasize - this is a difficult class. Given that the Ph.D. students are required to take 3 courses, I expect that you will spend at least ~3h a day working on this course. Here is some advice how to do it effectively and how to know whether you understand the material:
• Ask questions during the lecture. Almost of the time, if you have a question, others students also are confused. For those rare occasions that you are the only confused person, your question will allow other students to catch breath and refresh their notes. This is a graduate class - if I don’t hear any questions, I am going to assume that everything is clear and continue writing. And I can write fast.
• Talk to me during the office hours. If you want to talk to me but cannot come during the office hours (or the ofice hours are not enough), ask me to find some other time. I will try to make my best to accommodate you. (I am going to be less sympathetic for a request to meet outside of the office hours coming during the last week before midterm and from a student who have never talked to me before.) Talk to the TA.
• Make sure that you have all the required material. The readings contain strictly more material than the lecture. In the same time, although you are required to read all the assigned material, I find it difficult to imagine a situation that I would ask for a part of material that was not covered during the lecture or the problem set.