Economics 204 Summer/Fall 2011
Lecture 15Friday August 12, 2011
Second Order Linear Dierential Equations
Consider the second order dierential equation y = cy + by with b, c R.
Rewrite this as a rst order linear dierential equation in two variables:
y(t) =
Economics 204 Summer/Fall 2011
Lecture 13Wednesday August 10, 2011
Section 5.3. Fixed Point Theorems: Brouwers and Kakutanis
We have already studied xed points for the very special case of contraction mappings.
Here we study them for general functions as
Econ 204
Mathematical Tools for Economists
Summer/Fall 2011
Instructors
Chris Shannon
511 Evans Hall
cshannon@econ.berkeley.edu
643-7283
Office hours:
drop-in: 11:00 12:00 MTWThF
other times by appointment
Oleksiy Shvets
shvets@berkeley.edu
Ivan Balbuzano
Economics 204 Summer/Fall 2011
Lecture 14Thursday August 11, 2011
Dierential Equations
Existence and Uniqueness of Solutions
Denition 1 A dierential equation is an equation of the form
y (t) = F (y(t), t)
where F : U Rn and U is an open subset of Rn R.
An
Economics 204 Summer/Fall 2011
Lecture 12Tuesday August 9, 2011
Inverse and Implicit Function Theorems, and Generic Methods:
In this lecture we develop some of the most important concepts and tools for comparative
statics. In many problems we are interest
Economics 204 Summer/Fall 2011
Lecture 11Monday August 8, 2011
Sections 4.1-4.3 (Unied)
Denition 1 Let f : I R, where I R is an open interval. f is dierentiable at x I if
f(x + h) f(x)
=a
h0
h
lim
for some a R.
This is equivalent to
f(x + h) (f(x) + ah)
=
Economics 204 Summer/Fall 2011
Lecture 10Friday August 5, 2011
Diagonalization of Symmetric Real Matrices (from Handout)
Denition 1 Let
1 if i = j
0 if i = j
ij =
A basis V = cfw_v1 , . . . , vn of Rn is orthonormal if vi vj = ij .
In other words, a basi
Economics 204 Summer/Fall 2011
Lecture 8Wednesday August 3, 2011
Chapter 3. Linear Algebra
Section 3.1. Bases
Denition 1 Let X be a vector space over a eld F . A linear combination of x1, . . . , xn X
is a vector of the form
n
i xi where 1 , . . . , n F
y
Economics 204 Summer/Fall 2011
Lecture 9Thursday August 4, 2011
Section 3.3. Quotient Vector Spaces1
Given a vector space X over a eld F and a vector subspace W of X, dene an equivalence
relation by
x y x y W
Form a new vector space X/W : the set of eleme
Economics 204 Summer/Fall 2011
Lecture 7Tuesday August 2, 2011
Section 2.9. Connected Sets
Denition 1 Two sets A, B in a metric space are separated if
AB = AB =
A set in a metric space is connected if it cannot be written as the union of two nonempty
sep
Economics 204 Summer/Fall 2011
Lecture 6Monday August 1, 2011
Section 2.8. Compactness
Denition 1 A collection of sets
U = cfw_U :
in a metric space (X, d) is an open cover of A if U is open for all and
U A
Notice that may be nite, countably innite, or
Economics 204 Summer/Fall 2011
Lecture 4Thursday July 28, 2011
Section 2.4. Open and Closed Sets
Denition 1 Let (X, d) be a metric space. A set A X is open if
x A > 0 s.t. B (x) A
A set C X is closed if X \ C is open.
See Figure 1.
Example: (a, b) is open
Economics 204 Summer/Fall 2011
Lecture 5Friday July 29, 2011
Section 2.6 (cont.) Properties of Real Functions
Here we rst study properties of functions from R to R, making use of the additional
structure we have in R as opposed to general metric spaces.
L
Economics 204 Summer/Fall 2011
Lecture 3Wednesday July 27, 2011
Section 2.1. Metric Spaces and Normed Spaces
Here we seek to generalize notions of distance and length in Rn to abstract settings.
Denition 1 A metric space is a pair (X, d), where X is a set
Economics 204 Summer/Fall 2011
Lecture 2Tuesday July 26, 2011
Section 1.4. Cardinality (cont.)
Theorem 1 (Cantor) 2N , the set of all subsets of N, is not countable.
Proof: Suppose 2N is countable. Then there is a bijection f : N 2N . Let Am = f (m).
We c
Economics 204 Summer/Fall 2011
Lecture 1Monday July 25, 2011
Section 1.2. Methods of Proof
We begin by looking at the notion of proof. What is a proof? Proof has a formal
denition in mathematical logic, and a formal proof is long and unreadable. In practi
Lecture 15
T
| now derive some properties of fg(t) dW .
o t
T T
Theorem 15.1: |g(t)dW|:= f|g(t)|2dt.
o t 0 2
Proof: Let V (s) be defined by equation 14.2. By equation 14.5,
N
T
' 2._ 2
hm lIVN|2 (lllg(t)ll2dt-
Ns
T
By the definition of lg(t) dW,
o 1
T
m H
Lecture 10
Before returning to the theory of discrete time dynamic programming, I present an
example to give you a flavor for how the theory is used.
Example: (One sector optimal growth model) Consider a world with two commodities,
labor and a produced go
Lecture 3
Determinants
a a
I 11 12
Consider the rows of a 2x2 matrix A = as vectors In R2, say v and v . The
a a 1 2
21 22
matrix A is nonsingular or invertible if and only if these vectors are independent, which
means that they do not lie on the same li
Lecture 2
Linear Transformations
The next topic is functions from one vector space to another that preserve their linear
structure.
Definition: If A and B are non-empty sets, a function ffrom A to B assigns a single point
in B to every point in A. Such a