Math Camp 2019

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Banach-Tarskifield, an Axiom of Choice comic by Manyhills. This work is licensed under a CC BY-NC-SA 3.0 license by Square Root of Minus Garfield.

Instructors: Shaofei Jiang, Kriti Jain

Full syllabus: download here

Lecture notes:

Overview

The Math Camp introduces mathematical tools that are necessary for the first year PhD courses in Economics. Basic principles in analysis and optimization theory will be covered. This is also the first part of a two-course sequence on Math for Economics PhD students. ECO 385D (Math for Economists) will be offered in the Fall semester.

Textbooks

The following text will be followed:

    • An Introduction to Mathematical Analysis for Economics Theory and Econometrics, by Corbae, Stinchcombe and Zeman (Princeton University Press, 2009)
    • A First Course in Optimization Theory, by Sundaram (Cambridge University Press, 1996)

Additional useful text include:

    • Mathematics for Economists, by Simon and Blume (1994)
    • Microeconomic Theory, by Mas-Colell, Whinston and Green (Oxford University Press, 1995)
    • Recursive Methods in Economic Dynamics, by Stokey and Lucas (Harvard University Press, 1989)

Occasional lecture notes and readings will also be shared as the course proceeds.

Course outline

The following topics will be covered. The textbook chapters in the parentheses are only suggestive, and only a subset of topics in the textbooks will be covered.

  1. Preliminaries (CSZ Ch. 1-3)
    • Logic
    • Set theory
    • Binary relations (example: preference)
    • Tarski’s fixed point theorem
    • Binary operations, rings, fields, real numbers
  2. Static optimization (RS Ch. 2-6, Appendix of MWG)
    • Convexity (CSZ Ch. 5)
    • Separation Theorems and Unconstrained optimization
    • Constrained optimization (Lagrange multiplier method and Kuhn-Tucker method)
    • Quasi-Concavity in optimization
    • Supermodularity and monotone comparative statics
    • Theorem of Maxima and Envelope Theorem
  3. Topological and metric spaces (CSZ Ch. 4-6, 10)
    • Metric spaces
    • Topological spaces
    • Convergence
    • Continuity of functions and correspondences
    • Compactness
    • Fixed point theorems
  4. Normed linear spaces (CSZ Ch. 4-6, 10)
    • Normed linear spaces and Banach spaces
    • Bounded linear operators and dual spaces
    • Finite dimensional NLS
    • Lebesgue spaces
    • Hann-Banach theorem and separation theorems
    • Alaoglu theorem, weak and weak-* topology
    • Modes of convergence of random variables (if time permits)
  5. Dynamic programming (SL Ch. 1-5 (selected topics))
    • Introductory Macro examples and preliminaries
    • Deterministic Models (Both Finite and Infinite Horizon Problem)
    • Characterization of policy function
    • Applications of DPP under certainty
    • Continuous time problems (if time permits)

Download full syllabus here