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 wr
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
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
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 s
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 understandi
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
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 econ
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 Euclid
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
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
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. The
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 t
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)
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
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 wron
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 sop
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