BEEM103 Optimization Techiniques for Economists
Dieter Balkenborg
Departments of Economics
Lecture Week 1
University of Exeter
"Since the fabric of the universe is most perfect, and is the work of a most
perfect creator, nothing whatsoever takes place in

Dieter Balkenborg
BEEM103 Optimization Techiniques for Economists
Departments of Economics
University of Exeter
Homework Week 6
x
Exercise 1 Find the solutions of the form e
x
to the homogenous equation
x
_
6x = 0
Solution 1 The characteristic equation is

Dieter Balkenborg
BEEM103 Optimization Techniques for Economists
Departments of Economics
University of Exeter
Homework Week 1 Solutions
Exercise 1 Bring onto common denominator and simplify
2t t2
2t + 2
5t
t
2t
2
t
2
Solution 1
2t t2
2t + 2
5t
t
2t
2
t
=

BEEM103 Optimization Techniques for Economists
Dieter Balkenborg
Departments of Economics
University of Exeter
Homework Week 2 SOLUTIONS
Exercise 1 Calculate the rst and second order partial derivatives of
a) z
=
(2
x
y) x + 5 + 2x
b) z
=
x2 + y 3
(2
x
3y

BEEM103 Optimization Techniques for Economists
Dieter Balkenborg
Departments of Economics
University of Exeter
Homework Week 4 Solutions
Exercise 1 Let
be the triangle spanned by the points ( 1; 0), (1; 1) and (1; 1).
Sketch the triangle. Describe the tri

BEEM103 Optimization Techniques for Economists
Dieter Balkenborg
Departments of Economics
University of Exeter
Homework Week 3 SOLUTIONS
Exercise 1 Show that the function
Q=K L
with ;
> 0 and
+
< 1 is concave.
Solution 1 It is assmed here that K and L are

BEEM103
UNIVERSITY OF EXETER
BUSINESS School
January 2009
Sample exam Solutions, Part A
OPTIMIZATION TECHNIQUES
FOR ECONOMISTS
Part A
(You can gain no more than 55 marks on this part.)
Problem 1 (10 marks) Simplify
p p p
3
8 x2 4 y 1=z
p p p
2 3 x y5 z
So

BEEM103
January 2010
OPTIMIZATION TECHNIQUES FOR ECONOMISTS
solutions
Part A
(You can gain no more than 55 marks on this part.)
Problem 1 (10 marks) Simplify
p
1 x2 y 1
8 x3
2 1
p
3 x2 y
p
3
x= x2
and
7
x3
Solution 1
1
8
p
x2 y
x3
1
2 1
3 1
16 1
13 1
p =

BEEM103 Optimization Techniques for Economists
Dieter Balkenborg
Departments of Economics
University of Exeter
Class Exercises Week 3 - Solutions
Exercise 1 A monopolist produces two commodities in quantities Q1 and Q2 . The prices
for the two commodities

Dieter Balkenborg
BEEM103 Optimization Techniques for Economists
Departments of Economics
University of Exeter
Class Exercises Week 2 Solutions
Exercise 1 Calculate the partial derivatives of
a) z = 5y 5 + 4x4 y + 3x2 y 3 + 2xy 4 + 2x + 3y + 5
xy 2
b) z =

BEEM103 Optimization Techniques for Economists
Dieter Balkenborg
Departments of Economics
University of Exeter
Class Exercises Week 6 - Solutions
Exercise 1 Find the solutions of the form e
x
t
2x + x = 0
_
Solution 1 For such a solution x = e t , x =
_
x