Economics 511 Professor John H. Nachbar Fall 2002
Midterm
You have one and a half hours. Write your answers clearly, with good penmanship and good syntax. A "correct" but unintelligible answer is a wrong answer. 1. Prove the following. Theorem. Let (X, d)
Economics 511 Professor John H. Nachbar Fall 2004
Final
You have one and a half hours. Write your answers clearly, with good penmanship and good syntax. A correct but unintelligible answer is a wrong answer. 1. Prove the following. Theorem. The series xi
Econ 4111 Professor: John Nachbar January 24, 2008
Existence of Optima. 1 Introduction.
The mathematics of maximization is the mirror image of the mathematics of minimization: minimizing a function f is the same thing as maximizing the function f . Throug
Professor John Nachbar Econ 4111 January 29, 2008
Multivariate Dierentiation
1 Preliminaries
These notes provide an introduction to multivariate calculus. I assume that you are already familiar with standard concepts and results from univariate calculus.
Econ 511 Professor: John Nachbar November 30, 2008
Monotone Comparative Statics
1 Overview
Given an optimization problem indexed by some parameter , comparative statics seeks a qualitative understanding of how the solution changes with . If, for example,
Econ 4111 Professor: John Nachbar 11/20/08
Metric Spaces 1
1.1
Metric Spaces Basics.
Metric spaces.
A metric space (X, d) consists of a set of points, X together with a distance function, or metric, d : X X R. The interpretation is that d(a, b) is the dis
Econ 511 Professor: John Nachbar December 2009
Fixed Point Theorems
1 Overview
Denition 1. Given a set W and a function f : W W , x W is a xed point of f i f (x ) = x . Many existence problems in economics for example existence of competitive equilibrium
Economics 511 Professor: John Nachbar October 19, 2009
Basic Properties of the Euclidean Norm.
Given x RN , define x = (x x)1/2 .
If N = 1 then x = |x|, the absolute value of x. x is called the Euclidean norm of x. It measures the distance, in the standar
Econ 511 Professor: John Nachbar October 5, 2008
Continuity and Connectedness
Theorem 1. Let (X, dx ) and (Y, dY ) be metric spaces. Let f : X Y be continuous. Then for any connected set E X, f (E) is connected. Proof. I argue by contraposition. Suppose t
Professor John Nachbar Econ 4111 February 14, 2007
Finite Dimensional Optimization, Part II Second Order/Sufficient Conditions
1 Introduction
As noted in part I of these notes, the Kuhn-Tucker theorem only gives necessary conditions for a local maximum. E
Econ 4111 Professor: John Nachbar 9/25/08
Compactness and Completeness in RN . 1 R is complete.
Theorem 6, the Heine-Borel theorem, states that a set in RN is compact iff it is closed and bounded. Theorem 6 is immediate if I can show (a) that RN is comple
Econ 511 Professor: John Nachbar Fall 2008
Required Theorems for 511 Midterm
The following is a list of theorems that may appear as test questions. You should be able to prove any one of these. Note that some results are from the 4111 notes and some from
Economics 511 Professor: John Nachbar Fall 2008
Midterm Answers
5. Theorem. Let (X, dx ) and (Y, dy ) be metric spaces and suppose f : X Y and g : X y are both continuous. Let E X be dense in X . (a) If cfw_yt is a sequence in Y and yt converges both to
Economics 511 Professor: John Nachbar Fall 2008
511 Midterm
You have until 11:30AM. You can use either pen or pencil but write legibly and with good syntax. A "correct" but unintelligible answer is a wrong answer. Prove the following. 1. Theorem Let (X, d
Economics 511 Professor John H. Nachbar Fall 2004
Midterm
You have one and a half hours. Write your answers clearly, with good penmanship and good syntax. A correct but unintelligible answer is a wrong answer. 1. Prove the following. Theorem. Let (X, d) b
Economics 511 Professor John H. Nachbar Fall 2002
Midterm Answers
5. (a) Consider the sequence cfw_1/2, 1/4, 1/2, 3/4, 1/8, 1/4, 3/8, 1/2, . . . . The set of subsequential limits is [0, 1]. A more sophisticated approach is to note the following theorem, w
Economics 511 Professor John H. Nachbar Fall 2008
Final
You have two hours. Write your answers clearly, with good penmanship and good syntax. A "correct" but unintelligible answer is a wrong answer. 1. Prove the following. Theorem. Let (X, dx ) be a metri