Errata or Typos in book "An Introduction to Mathematical Analysis for Economic Theory and Econometrics"

Textbook  by Corbae, Stichcombe, and Zeman -CSZ hereafter.

typos: http://press.princeton.edu/releases/m8898_errata.pdf

Problem Set 2 - Set Theory (Applied Mathematical Methods to Economics - Summer Course)

PROBLEM SET 2 – PART 2 – SET THEORY 

Exercises from Corbae, Stinchcombe, and Zeman (2009) – odd ones

Exercise 2.2.5 (Notation and Other Basics)

Exercise 2.2.7 (Notation and Other Basics)

Exercise 2.2.13 (Products, Relations, Correspondences, and Functions)

Exercise 2.5.3 (Optimal Choice for Finite Sets)

Exercise 2.5.7 (Optimal Choice for Finite Sets)

Exercise 2.6.3 (Direct and Inverse Images, Compositions)

Exercise 2.6.5 (Direct and Inverse Images, Compositions)

Exercise 2.6.11 (Direct and Inverse Images, Compositions)

Exercise 2.6.19 (Direct and Inverse Images, Compositions)

Exercise 2.6.27 (Direct and Inverse Images, Compositions)

Exercise 2.7.9 (Weak and Partial Orders, Lattices)

Exercise 2.7.15 (Weak and Partial Orders, Lattices)

Exercise 2.8.3 (Monotonic Changes in Optima: Supermodularity and Lattices)

Exercise 2.8.11 (Monotonic Changes in Optima: Supermodularity and Lattices)

Exercise 2.8.13 (Monotonic Changes in Optima: Supermodularity and Lattices)

Exercise 2.8.15 (Monotonic Changes in Optima: Supermodularity and Lattices)

Exercise 2.8.19 (Monotonic Changes in Optima: Supermodularity and Lattices)

Exercise 2.8.21 (Monotonic Changes in Optima: Supermodularity and Lattices)

Exercise 2.9.3 (Tarski’s Lattice Fixed-Point Theorem and Stable Matchings)

Exercise 2.9.15 (Tarski’s Lattice Fixed-Point Theorem and Stable Matchings)

Exercise 2.9.19 (Tarski’s Lattice Fixed-Point Theorem and Stable Matchings)

Exercise 2.9.23 (Tarski’s Lattice Fixed-Point Theorem and Stable Matchings)

Exercise 2.9.27 (Tarski’s Lattice Fixed-Point Theorem and Stable Matchings)

Exercise 2.9.29 (Tarski’s Lattice Fixed-Point Theorem and Stable Matchings)

Exercise 2.10.3 (Finite and Infinte Sets)

Exercise 2.10.11 (Finite and Infinte Sets)

Exercise 2.10.13 (Finite and Infinte Sets)

Exercise 2.10.17 (Finite and Infinte Sets)

Exercise 2.12.7 (Revealed Preference and Rationalizability)

Exercise 2.13.2 (Superstructures)

Exercise 2.15.1 (End-Chapter Problems)

Exercise 2.15.3 (End-Chapter Problems)

RECOMMENDATIONS TO APPLIED REAL ANALYSIS

 Newest

BASS,, R.; Real Analysis for Graduate Students, Secod Edition; CreateSpace Independent Publishing Platform, 418p. 2013

                        Classics

APOSTOL, T.; Mathematical Analysis, Pearson, 492p., 1974

BARTLE, R.; Sherbert, D.; Introduction to Real Analysis; Wiley, 416p., 2011

GARDING, L.; Encounter with mathematics, Springer-Verlag, 270p., 1977

KOLMOGOROV, A.; FOMIN, S.; SILVERMAN, R.; Introdutory Real Analysis, Dover Publications, 416p., 1975

RUDIN, W.; Principles of Mathematical Analysis; McGraw-Hill, 325p., 1976

WITTGENSTEIN, L.; Tractatus Logico-Philosophicus, CreateSpace Independent Publishing Plataform, 96p., 2011

For Brazilians: challenging, intensive and provocative

LIMA, E.; Análise Real – volume 1: Funções de uma Variável, 11ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Matemática Universitária, 198p., 2012

LIMA, E.; Análise Real – volume 2: Funções n Variáveis, 5ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Matemática Universitária, 209p., 2010

LIMA, E.; Curso de Análise vol. 1, 14ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Projeto Euclides, 431p., 2013

LIMA, E.; Curso de Análise vol. 2, 11ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Projeto Euclides, 547p., 2012

                       Applied Real Analysis

OK, E.; Real Analysis with Economic Applications; Princeton University Press, 664p. 2007

(We intend to use this book on 2015 Summer Course)

Questions:

The aim of this first chapter is to provide the necessary elements for understanding the structure of theoretical models in economics. In a point of view, study the models and their detailed assumptions. So, based in a framework we could test, corroborate and infer the theoretical structure thereof.  

Questions:

What’s the differences amongst theorem, lemma, corollary, and axiom?

Why there exists?

Why we need to prove them or take some of them as truth?

Theorem: a theorem in a theoretical model is a statement that has been proven on the basis of theoretical foundations and previous concepts, such as other theorems, corollaries, axioms, and lemmas. For instance, the Nash Theorem.

Lemma: is a “helping theorem”, a proposition with little applicability except to be a part of the proof of a larger theorem. In some cases, as in Finance  we could cite the Ito’s Lemma.

Corollary: is a proposition that follows with little or no proof from one other theorem or definition. Example: Walras’ Corollary in General Equilibrium.

Axiom: considerate a postulate is a statement that is accepted without proof and regarded as fundamental to a subject. The main example is the Probability Axioms.

Note: one of the most interesting articles that shows the importance of ability in mathematics was written by Franklin Joel, entitled “Mathematical Method of Economics”, published by The American Mathematical Monthly in 1983. In that paper, the author emphasizes the role of mathematics in academic work of prized economists.

            It’s also interesting to stress that, in the nineteenth century, a scholar or noble person was recognized for his skills in chess and mathematics. Seeing mathematics as a major obstacle to Lord Keynes, in the twentieth century, he wrote to another economist complaining about studies in math.

           

Problem Set 1 - Logic (Applied Mathematical Methods to Economics- Summer Course)

APPLIED MATHEMATICAL METHODS TO ECONOMICS

One of the most important characteristics of an economist is its ability to abstract from reality in a complex environment and endless relationships amongst  actors, institutions, households and time. Therefore, quantitative methods help you to model or caricature the complex reality in models that express events forming consolidated theories. Furthermore, it allows quantifying the relations theories formalized taking as a starting point. Based on this assertion mathematics has become an important tool for the economist and evaluate this summer part of the book entitled entitled "An Introduction to Mathematical Analysis for Economic Theory and Econometrics, by Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman, Princeton University Press, 2009.

ECO 110 -  APPLIED MATHEMATICAL METHODS  TO ECONOMICS

A – SYLLABUS

  

Part 1 – Logic: Statements, Sets, Subsets and Implications; Statements and Their Truth Values; Proofs, a First Look; Logical Quantifiers; and, Taxonomy of Proofs.

References:  Corbae, Stinchcombe e Zeman (2009, chap. 1)

Part 2 – Set Theory: Products, Relations, Correspondences and Functions; Equivalence Relations; Optimal Choice for finite Sets; Direct and Inverse Images, Compositions; Weak and Partial Orders, Lattices; Monotonic Changes in Optima: Supermodularity and Lattices; Tarski’s Lattice Fixed-Point Theorem and Stable Matchings;  Finite and Infinite Sets; The Axiom of Choice and Some Equivalent Results; Revealed Preference and Rationalizability; and,  Superstructures;

References: Corbae, Stinchcombe e Zeman (2009, chap. 2) e Elon (2012, chap.1)

Part 3 – The Space of Real Numbers: Basic Properties of Rationals; Distance, Cauchy Sequences, and the Real Numbers; The Completeness of the Real Numbers; and, Supremum and Infimum;

References: Corbae, Stinchcombe e Zeman (2009, chap. 3) e Elon (2012, chap. 2)

Part 4 – The Finite-Dimensional Meric Space of Real Vectors:   The Basic Definitios for Metric Spaces; Discrete Spaces;  as a Normed Vector Space; Completeness; Closure, Convergence; and Completeness; Separability; Compactness in ; Continuous Function in   ; Lipschitz and Uniform Continuity; Correspondences ant the Theorem of the Maximum; Banach’s Contraction Mapping Theorem; and, Connectedness;

References: Corbae, Stinchcombe e Zeman (2009, chap. 4)

Part 5 – Finite-Dimensional Convex Analysis:    The Basic Geometry of Convexity; The Dual Space of ;  The three Degrees of Convex Separation; Strong Separation and Neoclassical Duality; Boundary Issues; Concave and Convex Functions;; Separations and the Hahn-Banach Theorem; Separation and the Karush-Kuhn-Tucker Theorem[1]; Interpreting Lagrange Multipliers; Differentiability and Concavity; Fixed-Point Theorems and General Equilibrium Theory; and, Fixed-Point Theorem for Nash Equilibria and Perfect Equilibria;

References: Corbae, Stinchcombe e Zeman (2009, chap. 5)

Part 6 – Metric Spaces: The Space of Compact Sets and the Theorem of the Maximum; Space of Continuous Functions; , the Space of Cumulative Distribution Functions; Approximation in    when    is compact; Regression Analysis as Approximation Theory; Countable Product Spaces and Sequence Spaces; Defining Functions Implicitly and by Estimation; The Metric Completion Theorem; and, The Lebesgue Measure Space;

References: Corbae, Stinchcombe e Zeman (2009, chap. 6)

B – REFERENCES

CORBAE, D.; STINCHCOMBE, M.; ZEMAN, J.; An Introduction to Mathematical Analysis for Economic Theory and Econometrics, Princeton: Princeton University Press, 671p., 2009.

LIMA, E.; Análise Real – volume 1: Funções de uma Variável, 11ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, 198p., 2012

C- GRADES 

Three exams as follows: 

(i)                 20%  - itens 1 and 2;   February 07

(ii)               30% -  itens 1, 2, 3, and 4 (cumulative);  February 26, and

(iii)             50% -  itens 1, 2, 3, 4, 5, and 6 (cumulativE);   March 19.

            The candidates could follow notes and exercises in this blog:

                         http://mathematical-economics.blogspot.com.br/

---

[1] We recognize Karush’s contribution on Theorem.