August 20, 2013 Truman Bewley
Math Camp
Homework #14
1) A measure on the Borel subsets Q3 of R is a function u: 93 a [0, co) such that u( o) = 0 , and
no
if A1, A2, is a sequence of mutually disjoint sets in 93, then u[ o A ) = i u( A) . Lebesgue
n=1 n
n=
Lecture 9
Proof of the KuhnTucker Theorem
The KuhnTucker theorem is so central to economic thinking that its proof is worth
understanding. It also provides the occasion to introduce new concepts that are widely used in
economic theory. I first prove the
Lecture 4
Real Analysis
Real analysis starts from the simple idea of closeness to each other of numbers and of N
vectors of numbers. The set of points close to a given point is taken to be points in an
Ndimensional ball of small radius about the point.
Lecture 5
Correspondences and Berge’s Maximum Theorem
l present Berge’s maximum theorem, which provides conditions under which the
maximum value and the maximizing vectors in a constrained maximization problem depend
continuously on parameters in the desc
Lecture 1
Introduction
For those of you who have previously studied the material presented in these lectures, I
hope it will revive and organize old memories. For those who have not seen much of the
material, there is a risk that the course will be overwh
Appendix to Lecture 5, Discussion 0f the Fundamental Theorem of Calculus
l here deal with the question of why the integrand f is assumed to be continuous in the
fundamental theorem of calculus. We must distinguish two fundamental theorems of calculus.
The
Appendix to Lecture 5, Proofs
b
 now show that the limit appearing in the definition of the Riemann integral {f(y) dy
exists, when f is a continuous function. First of all, I need a definition and a lemma.
Definition: If f : A a B, where A is a subset of
Lecture 6
Multivariable Calculus
Multivariate like univariate calculus has to do with the local approximation of functions
by affine ones.
Definition: Let U be an open subset of Ft” and let f: U a R“. The function f is
differentiable at c e U if there exi
Appendix to Lecture 11, Proofs
In this appendix I provide proofs not covered in class for lack of time.»
Lemma 11.12: Suppose that assumptions 5 and 8 apply. lf 1 e X andy e int (3(5) ,
then there is a positive number 5 such that y 6 int G(x) , if x — :H
Appendix to Lecture 9, Example
Example: A consumer accumulates money balances, has a nonrandom but fluctuating
and exogenously determined income, and in every period chooses how much to spend on
consumption. The consumer’s choice variable in period t is
Lecture 7
Taylor’s Theorem
Taylor’s Theorem 7.1: Let f: U a R, where U is an open subset of RN. Suppose that U
contains the straight line segment from a to b, where a ¢ b, and suppose that f is r times
differentiable, where r 2 1. Then there exists a poin
Lecture 10
I present a specific example of the growth model discussed at the end of lecture 9.
Example 10.1: Let y = F( K) : 2 K, v(C) = n(C), and 0 < (3 <1.
The objective is to maximize iB'n( C) subject to C + K s 2 K 1 , fort: 0, 1, 2, , where
l t [—
Appendix to Lecture 12, Example
l present an example to illustrate how optimal control theory is used in economic
models.
Example (A Simple thimal Economic Growth Model): Imagine an economy in which
there is one commodity that may be used as capital or fo
Appendix to Lecture 11, Examples
The following example shows that the value function V may not be continuous if we do not
assume that G is lower semicontinuous.
Example: (Discontinuous value function) Let
Xzﬁygﬁ)€RH(w3V+lg—3Vs2t
As a step in the definiti
Lecture 13
I now show the connection between the Bellman equation and_the Hamﬂtpn Jacobi Bellman
equation by deriving the latter from the former. Let the functions x (t) and u (t) solve of the
problem
max TJf(x(t), u(t)dt
u 54 0
s.t. dX(t) = g(x(t), u(t
Lecture 12
Optimal Control Theory
Imagine that you control an allterrain vehicle that must go from point x to point x .
0 1
There are varied types of ground between x0 and x1, hilly, flat, sandy, marshy, grassy, and so
on. The objective is to choose the
Lecture 3
Determinants
Definition: An NxN matrix A is non—singular if the equation Ax = 0 has no nonzero solution.
Remark: By theorem 2.6, an NxN matrix is nonsingular if and only it it is invertible.
aa
1112
Consider the rows of a 2x2 matrix A = as vec
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 nonempty sets, a function f from A to B assigns a single point
f(a) in B to every point a in A.
August 2, 2013 Truman Bewley
MLCamb
Homework #2
Suppose that there are N commodities that are produced from each other and labor
according to a linear technology. Suppose that in order to produce one unit of commodity n, for
n = 1, N, b units of labor are
August 13, 2013 Truman Bewley
Math Camp
Homework #9
There are N produced goods and K primary inputs. Production of one unit of the nth
produced good requires a units of the mth produced good and b units of the kth primary
mn kn
input, where a 2 O and bk >
August 15, 2013 Truman Bewley
Math Camp
Homework #1 1
A consumer has the following utility function for consumption expenditure in each period:
U(C) : C, ”0303100
100, ifC> 100.
The consumer has wealth of y at the beginning of period t, for t = O, 1, . Th
August 13, 2013 Truman Bewley
mm
Homework #10
There is a fish farm in which fish are started in period 1 and may be harvested in each of
T > 1 successive periods, t = 1, . , T. The question is how much to harvest in each period. The
stock of fish at the b
August 6, 2013 Truman Bewley
Math Camp
Homework #4
Suppose that good 1 is produced from good 2 according to the production function
y : f (y ), where y11 is the output of good 1 and y is the nonnegative input of good 2 in the
11 1 12 12
production of goo
August 1, 2013 Truman Bewley
Math Camp
Homework #1
1) Consider the following matrix.
19 8 —21422
63—247
72268
14421
45—1005
a) Put this matrix in row reduced echelon form.
b) How many linearly independent columns does this matrix have?
c) How many linearl
August 8, 2013 Truman Bewley
Math Camp
Homework #6
1) Consider the function F: R a R defined by the equation
F(t) : tZSln(t_1), ift¢0
0, if t = O.
Show that F is everywhere differentiable and that the derivative of F is not continuous at O.
2) Let u : RN
August 5, 2012 Truman Bewley
Math Camp
Homework #3
1) Let W be the linear span in R5 of the following vectors:
V =(11251aoto)!
1
v2=(0,1,3,3,1),and
v =(1,4,6,4,1).
3
Find a basis for Wt.
2) Find the inverse of the matrix
123
312
231
3) N observations are
August 8, 2013 Truman Bewley
Math Camp
Homework #7
1) A consumer has an income of 162 and buys quantities of three commodities, goods 1, 2, and
3. The price of good 1 is 1, that of good 2 is 2, and that of good 3 is 3. The consumer’s utility
function is

August 7, 2013 Truman Bewley
Math Camp
Homework #5
1) This problem has to do with the seeming paradox that in some types of businesses, such as
fast food restaurants, an increase in the minimum wage can increase employment.  start with a
numerical exampl
August 12, 2013 \ Truman Bewley
Math Camp
Homework #8
1) Suppose that a consumer buys two commodities and that the consumer’s relative interest in
the first good increases with its relative price. More specifically, assume that the utility
function of the