Maxwell B Stinchcombe

OPTIMAL USE OF EXHAUSTIBLE AND RENEWABLE RESOURCES; VALUATION OF NONMARK ETED ENVIR AMENITIES; ECON OF POLLUTION CONTROL INSTRUMENTS; ENVIR QUALI TY AND INTNATL TRADE; ECON OF GLOBAL CLIMATE CHANGE; POLLUTION CONTROL.

PREREQUISITE: ECO 420K AND 329 WITH A GRADE OF AT LEAST C- IN EACH.

This course will study the economics of public policy toward natural resources and the environment. It is designed primarily for advanced undergraduates in economics. Prerequisites include microeconomics and calculus. We will start with the concepts of externalities, public goods, property rights, market failure, and social cost-benefit analysis. Within this framework, we will consider a few additional problems such as information, uncertainty, and risk analysis. The first set of applications of these tools will involve natural resources. Other applications include air pollution, water pollution, solid waste management, and hazardous substances. In addressing each of these problems, we will compare public policy responses such as administrative regulation, marketable permits, tax incentives, and direct subsidies.

SYLLABUS FOR MATHEMATICS FOR ECONOMISTS

ECONOMICS 362M, SPRING 2010

MAXWELL B. STINCHCOMBE

Organization

Basics: We meet Tuesdays and Thursdays, from 12:30 p.m. to 2:00 p.m. in UTC

3.122. The unique number is 33720.

Reaching me: My office is BRB 2.118, phone number (512)475-8515, my e-mail

address is max.stinchcombe@gmail.com.

Office Hours: Tuesdays and Thursdays, 2:00-3:00 p.m. and Mondays and Wednesdays, 11-11:45 a.m., though the graduate students have priority at that time. You can also drop-in, and/or make an appointment.

Teaching Assistant: We are lucky to have Mohammad Dehghani as the T.A. for this course. His office hours are Mondays and Fridays, 2-3:30 p.m. His e-mail is

mhdehghani@gmail.com, his phone is 512-767-9625, and his office is BRB 2.146.

Texts: The required textbook for this class is An Introduction to Mathematical

Analysis for Economic Theory and Econometrics by D. Corbae, M. Stinchcombe, and

J. Zeeman.  I strongly advise you to read the material once before we cover it in lecture and once afterward.

Evaluation: There will be weekly homework assignments (40%), consisting of about

8 problems of varying difficulties, two mid-terms, Thursday February 25, 2010 (10%)

and Thursday March 25, 2010 (10%), and annual exam, Saturday May 15, 2010,

9a.m. - 12 p.m. (40%).

In more detail:

(1) The homework assignments are due every Tuesday beginning January 26, unless this falls during spring break (March 15-20), in which case the assignment is due the following Tuesday.

(a) I expect you to get almost all of the homework problems correct. This is because you will have two chances at them. From the time I hand an assignment back to you, you have two weeks to correct any errors on the homework problems. I am very happy to talk about any difficulties you have with the assignments.

(b) I strongly encourage you to work together on assignments, and to talk to me about any and all parts of the assignments. However, I strongly discourage you from merely copying someone else's work. If you \free ride" on others' efforts, it is likely that you will not understand what is being taught sufficiently well to pass the class.

(2) The two mid-terms will each count for 10% of your grade, and they will likely include both an in-class and a homework-like component. They will be given on Thursday February 25 and Thursday March 25. I do not give make-up exams.  If you must miss the mid-term on University business, make arrangements with me in advance.

(3) The final, cumulative exam will be given on Saturday May 15, 2010, from 9a.m. till 12 p.m. It will count for the remaining 40% of your grade.

Background: I will assume that you have had a good introduction to microeconomics, especially the utility maximization foundation of the theory of consumer choice and the theory of the firm, and that you have a good working knowledge of partial derivatives and differential calculus.

Topics: The course is in introduction to the basics mathematics that is used in essentially all advanced economic analysis. Though it may feel that the tools are central to your experience of this course, it is their usefulness that motivates me to teach this course, and I hope you come to appreciate these as well. We will see these tools used for

(1) Rational choice theory, which is the basic model of human motivation that economists use.

(2) Monotone comparative statics, which examines how optima move as circumstances change.

(3) One factor growth models, which are the basic dynamic models for resource economics and macroeconomics.

(4) Duality theory, which ties together the different approaches to consumer demand theory and to the behavior of price-taking firms.

(5) Constrained optimization theory and the Karesh-Kuhn-Tucker Theorem, which is the basic tool for analyzing the mathematical model of human behavior that economists use.

Schedule:

The intended schedule is given below. If needed, we will adjust it.

A COURSE ON INFINITE GAMES

MAXWELL B. STINCHCOMBE

Contents

1. Introduction 3

1.1. Organization 3

1.2. Background 3

1.3. Outline of Topics 3

1.3.1. Finite Games and The Equilibrium Correspondences 4

1.3.2. Compact and Continuous Games 4

1.3.3. In nitely Repeated Games 4

1.3.4. Discontinuous Normal Form Games 4

1.3.5. In nite Extensive Form Games 4

2. Finite Normal Form Games 6

2.1. Interpretation and Equilibria 6

2.2. Properties of the Equilibrium Correspondence 7

3. Ideal Points and In nitesimals 9

3.1. Adding In nitesimals to R 9

3.2. Constructing R 10

3.3. In nitesimals and Standard Parts 11

3.4. Continuity via In nitesimals 13

3.5. Compactness via the Standard Part Mapping 14

3.6. The Space of Compact Sets 15

3.7. Monads 16

3.8. Continuity of Correspondences 17

3.9. Appendix 19

3.9.1. Internal Sets 19

3.9.2. Star Finite Maximization 21

3.9.3. Perfection and Properness 22

3.9.4. The Closed Limsup and Liminf 23

4. The Equilibrium Correspondences for Finite Games 24

Date: Fall 2009.

1

2 MAXWELL B. STINCHCOMBE

5. Compact and Continuous Games 27

5.1. Equilibrium Existence 27

5.2. Perfection and Properness 27

5.3. Nearly Compact and Continuous Games 27

5.4. Extensive Form Games 27

5.4.1. In nitely Repetitions of Finite Stage Games 27

6. Compact and Discontinuous Games 28

6.1. Examples 28

6.2. Adding Ideal Points 28

6.3. Games in Which Ideal Points Are Not Needed 28

7. Extensive Form Games 29

References 30

A COURSE ON INFINITE GAMES 3

1. Introduction

The aim of this course is to take you from Nash's equilibrium exis-

tence theorem for nite normal form games up to the limits of what is

known about equilibria in extensive form games in which pure action

spaces are in nite. Combined with a review the basic mathematics of

probabilities on well-behaved metric spaces and nets of nite approx-

imations, the rst half of the course will cover in nite normal form

games. Building on the material in the rst half, the second half of the

course will cover in nite extensive form games.

1.1. Organization. Meeting: We will meet Mondays and Wednes-

days, 2 to 3:30 p.m. in BRB 1.120. We will also have a weekly problem

session meeting at a time accomodating the largest number of people.

Workload: Throughout this set of notes are live links to the papers

we will be discussing. Lectures will be far more informative if you look

at the articles before class meetings.

For each of the topics there will be a homework that you will have

unlimited chances to hand in completely correct. A crucial skill that I

want you to develop is knowing when you have given a correct proof.

To ease this, we will have a weekly problem session meeting. If you skip

the problem session, I expect it to be because you have no problems

with the material.

Over the course of the semester, we will look at a number of re-

sults with applications to problems you care about. We will also see

a number of open theory problems. You should take this class as an

opportunity to work on these problems in paper form.

1.2. Background. As to economics, I expect you to have taken the

graduate introductory in microeconomics sequence and an introductory

course in game theory. As to mathematical tools, I expect you to have

had or to be willing to pick up basic real analysis at the level of a

good undergraduate course, e.g. the material in Chapter 4 in Corbae,

Stinchcombe, and Zeeman [3]. As we need it, we will cover parts of the

material on metric spaces, [3, Ch. 6], probabilities on metric spaces [3,

Ch. 9], and ideal points representing sequences [3, Ch. 11].

1.3. Outline of Topics.

4 MAXWELL B. STINCHCOMBE

1.3.1. Finite Games and The Equilibrium Correspondences. For the

rst two weeks, we will cover the basic properties of the equilibrium and

approximate equilibrium correspondences for nite games, introducing

ideal points representing sequences for the proofs. The material on

upper hemicontinuous and continuous correspondences in [3, Ch. 6]

will appear, as well as the rst section of [3, Ch. 11]. We will also, for

the rst time, see Fudenberg and Levine's [4] \most utility di
erence

it can make to anyone" metric on strategies.

1.3.2. Compact and Continuous Games. The subsequent two weeks

will be spent on ideal point representations of sequences of nite ap-

proximations to games with compact spaces of actions and jointly

continuous utility functions. The nite approximation approach to

equilibrium existence is in Glicksberg [5]. We will also look at the -

nite approximation approach to equilibrium re nement in Simon and

Stinchcombe [10], and Harris, Stinchcombe and Zame's [6] study of the

largest classes of games for which the compact and continuous approach

'works.'

1.3.3. In nitely Repeated Games. The most heavily studied class of

compact and continuous games are the in nitely repeated nite games.

Recent work has concentrated on games of incomplete monitoring. We

will look at two sets of results on the recursive structures in these

games, Abreu, Pearce and Stacchetti [1] and Amarante [2].

1.3.4. Discontinuous Normal Form Games. When compact normal form

games have discontinuous payo
s, there are two approaches. The rst

is to nd conditions on the games implying that equilibria still exist.

The second is to expand the notion of equilibrium by allowing correla-

tion devices of various kinds.

Within the rst approach, we will study Reny's [9] uni cation and

generalization of a long and dicult literature.

Within the second approach, the basic approach is in Simon and

Zame's [11] sharing rule equilibria. With the bene t of hindsight, their

sharing rule equilibria are not equilibria, and often not easily inter-

pretable, so we will work from Stinchcombe's [12] version of their idea.

1.3.5. In nite Extensive Form Games. The addition of correlation de-

vices pre-dated the study of its use in normal form games. The rst

breakthrough came in the study of signaling games, in which we have

A COURSE ON INFINITE GAMES 5

Manelli's work on the introduction of cheap talk. We will then turn to

the general study of extensive form games and the addition of commu-

nication as in Jackson et. al. [7], and in auctions Jackson and Swinkels

[8].

6 MAXWELL B. STINCHCOMBE

2. Finite Normal Form Games

A game is a collection

INTRODUCTION TO THE FORMAL STUDY OF INTERDEPENDENT DECISION MAKING. APPLICATIONS OF GAME THEORY INCLUDE PRICING AND ADVERTISING STRATEGIES, LABOR-MANAGEMENT BARGAINING, AND TARIFF NEGOTIATIONS.

PREREQUISITE: ECONOMICS 420K AND 329 WITH A GRADE OF AT LEAST C- IN EACH.

Contact professor for more information.

Prerequisistes: Eco 420k with at least a C- or better and M408D with at least a C- or better.

The course is in introduction to the basics mathematics that is used in essentially all advanced economic analysis. Though it may feel that the tools are central to your experience of this course, it is their usefulness that motivates me to teach this course, and I hope you come to appreciate these as well. We will see these tools used for:

(1) Rational choice theory, which is the basic model of human motivation that economists use.

(2) Monotone comparative statics, which examines how optima move as circumstances change.

(3) One factor growth models, which are the basic dynamic models for resource economics and macroeconomics.

(4) Duality theory, which ties together the different approaches to consumer demand theory and to the behavior of price-taking firms.

(5) Constrained optimization theory and the Karesh-Kuhn-Tucker Theorem, which is the basic tool for analyzing the mathematical model of human behavior that economists use.