ECON 505 LECTURE NOTES
LN #1 Logic, sets, functions
[Cf. Simon and Blume, Sections 2.1-2.2]
Logic
In the following,
The sentence not
The sentence
p
p
and
q
are sentences;
(also written as ~
(also written as
p&q
p
p
P
is a property.
) is true if and only i
ECN 505
HOMEWORK 2
1.
How many local maxima or minima and how many global maxima or minima does
the function
2.
f ( x ) 1/ 2 x 2 1 / 3 x3 3 / 2 x 4
have? Where are they?
Suppose two functions f and g both with domain [0,1] each have a strong global maximu
ECN 505
HOMEWORK 5
1. If
f
and
g
are both continuous at
x a,
must
g of
be continuous? If so, prove the
statement. Otherwise, present a counterexample.
2. If and
are both continuous, must be continuous? If so, prove the statement.
g
f
g of
Otherwise, prese
ECN 505
HOMEWORK 4
1. Prove by mathematical induction:
a) For every integer
,
n 1
b) For every integer
c) For every integer
n 1
n
n(n 1)
i 2
i 1
n
,
2
i3 i
i 1
i 1
n
n
.
.
, positive, negative, or zero,
2. Use the information in the class notes to find
1
ECONOMICS 505
MATHEMATICAL ECONOMICS
Fall 2015
Professor:
Jan Ondrich
Office:
426 Eggers Hall
Phone:
x-9052
Office Hours:
TTh 11-12 a.m. 3:30-4pmandbyappointment.
Email:
jondrich@maxwell.syr.edu
Texts:
Required
Carl P. Simon and Lawrence Blume. Mathemat
ECN 505
Problem Set 7
Professor Ondrich
1. The cost function of a firm is the minimum cost of producing a specific level of output,
you can call it Q0 , although any subscript is usually dropped and the cost function written
as C (Q, r1 , . . . , rn ) , w
ECON 505 LECTURE NOTES
LN #1 Logic, sets, functions
[Cf. Simon and Blume, Sections 2.1-2.2]
Logic
In the following,
The sentence not
The sentence
p
p
and
q
are sentences;
(also written as ~
(also written as
p&q
p
p
P
is a property.
) is true if and only i
ECN 505 LECTURE NOTES
LN #5: Many Variables
[Cf. Simon and Blume, Chapters 10, 12, 13, 14]
Optimization of functions of many variables, i.e., more than one.
Euclidean space:
vector:
R
( x1 , x2 , . . . , xn )
where
xi R
for all i.
Euclidean n-space:
n
R c
ECN 505 LECTURE NOTES
LN #2 Foundations of Optimization
Optima
We are interested in functions
f : A R
because
maximizing and minimizing. The order properties on
[Note that
1 2
R
R
has order properties that facilitate discussing
are the usual properties of
ECN 505 LECTURE NOTES
LN #6
Matrices & Determinants
[Cf Simon and Blume, Chapters 6,7,9]
A matrix is a (finite) rectangular array of real numbers, usually framed by square brackets:
1
5
M
0
14
3
1/ 71 1.05 257
.
The matrix M has 3 rows and 5 columns.
a
ECN 505 LECTURE NOTES
LN #4: Economic Applications
[Cf. Simon and Blume, Section 3.6]
1. The marginal as derivative.
If
y f x
Now,
where f is differentiable, marginal y is
AC C (q) / q
on
R
. Assume
C (q)
Dx y
.
is differentiable. We want to find an
inte
ECN 505 LECTURE NOTES
LN #3: Limits & Derivatives
[Cf. Simon and Blume, 2.3-2.6, Chaps 4, 5]
Essential point: If a "smooth" function
interior maximum at
at
x0
x0 A
f : A R
with
has a weak local
AR
then it occurs where there is a horizontal tangent line to
Take the integral:
x
7d):
fV3+2x
. x .
For the integrand — , substitute u = 2 x+ 3 and cite = 2(1):
V’2x+3
lfu—Sd
=— u
2 2V?
u—3 . V'u 3
,_ gwes— —
2 V“ n 2 2 V“:
=l ﬂ 3 ]du
Expanding the integrand
2 224K
Integrate the sum term by term and f
ECN 505
HOMEWORK 5
1. Find the partial derivatives of
g ( x, y , z ) 4 x 2 y 2 z 2 .
2. Find the partial derivatives of
y st 7
z x y
.
3. Find the product C of the two matrices
0 1 3 3
A
2 3 1 4
and
1
2
B
2
3
2
1
2
2
.
with respect to s and t, where
x
ECN 505
HOMEWORK 3
1.
Show with examples that even if f and g are both continuous at
xa
, that
f og
may not be continuous at a.
2.
If g and
3.
Is the function
f og
are both continuous, must f be continuous?
1/ x
continuous? Explain. (This is a trick quest
ECN 505
HOMEWORK 8
Professor Ondrich
1. The cost function of a firm is the minimum cost of producing a specific level of output,
you can call it Q0 , although any subscript is usually dropped and the cost function written
as C (Q, r1 , . . . , rn ) , wher
ECN 505
HOMEWORK 2
1.
How many local maxima or minima and how many global maxima or minima does
the function
2.
f ( x) 1/ 2 x 2 1 / 3 x3 3 / 2 x 4
have? Where are they?
Suppose two functions f and g both with domain [0,1] each have a strong global maximum
ECN 505
Problem Set 9
Professor Ondrich
1. A consumer has utility function U ( x, y ) xy and faces the budget constraint
2 x y 100 . Use the method of Lagrange Multipliers to find the optimizing values of x
and y . Find the marginal utility of income and
ECN 505
HOMEWORK 6
1. Suppose
where
h
A
is an
nxn
is similarly
nonsingular matrix and
n x1
. Show that
x A 1h
x
is an
n x1
vector satisfying
Ax h
,
.
2. Find the rank of the following matrices and if a matrix is nonsingular find its inverse.
Connect the r
ECN 505
HOMEWORK 7
1. Find the determinant of the following matrices
i)
1 2 2 3
1 0 2 0
3 1 1 2
4 3 0 2
ii)
2 1 3 2
3 0 1 2
1 1 4 3
2 2 1 1
2. For the following matrices, find the determinant and adjoint. Then use these to find the
inverse.
i
ECN 505
Problem Set 3
Answer to Final Problem
Even though it appears that f ( x ) 1/ x is discontinuous at x 0 , we must remember that 0
is not in the domain of f .
For any c R \cfw_0 , define
B (c) cfw_z :| c z | .
For any z B (c ), there exists v R such
ECN 505
HOMEWORK 1
1. Prove DeMorgans Laws:
( A B) c A c B c
; and
( A B) c A c B c
2. Prove that
( A [ B (C D c ) c ) c A c ( B c C c ) ( B c D )] c
3. Draw the following in
a)
b)
4. Draw in
.
R2
R2
.
[0,1) 1, 2
.
( 0 1, 2 ) ( 0,1 1, 2 )
the graph of the
/
ECN 505 LECTURE NOTES
LN #5: Many Variables
[Cf. Simon and ~lume, Chapters 10, 12, 13, 14]
Optimization of functions of many variables, i.e., more than one.
Euclidean space: R
vector: (x 1,x2 ,
,xJ where x; ER for all i.
Euclidean n-space:
Rn= cfw_(Xi,X
ECN 505 LECTURE NOTES
LN #9: Inequality Constraints
[Cf. Simon and Blume, 18.3-18.7]
Optimization with inequality constraints:
The Karush-Kuhn-Tucker approach.
Non-negativity constraints
Min C rK wL
Notice some
Q
subject to
,
,
K 0 L 0 ( K 1)( L 1) Q
can
ECN 505
HOMEWORK 4
1. Prove by mathematical induction:
a) For every integer
,
n 1
b) For every integer
n 1
n(n 1)
i
2
i 1
n
,
n
i
i 1
c) For every integer
n
3
2
.
.
i
i 1
n
, positive, negative, or zero,
2. Use the information in the class notes to fin
Problem Set 9
1. We are trying to maximize the transmission rate of a multi-carrier communication system
with N channels. Each carrier/channel can carry a signal power
under noise
pi 0
ni 0
ni 0
. The total power must be smaller or equal than P. The trans
ECN 505 LECTURE NOTES
LN #8: Equality constraints
[Cf. Simon and Blume, 18.1-18.2, Chapter 19.]
Optimization with equality constraints: The Lagrangian approach.
Comparison of two approaches to equality constraints:
Example 1. Maximize
V x y
subject to
x2