Homework 8
Mathematics for Economics
Textbook Questions:
Section 13.2 #6 (assume 0 and 0) and #8 in page 461
Section 13.4 #2 in page 472
Section 13.7 #5 in page 486: This question is challenging but w
Mathematics for Economics
Homework 3
Do the following problems in the Textbook.
Section 7.2 #2 and #4 in p. 213: For d2 Y /dI 2 in #4 (b), you need to use chain rule (since
Y is a function of I) based
Mathematics for Economics
Solution to Homework 3
Section 7.2 #2 and #4 in p. 213
Section 12.2 #7 and #9 in p. 411
Section 12.3 #4 in p. 416
1. Use the chain rule:
dZ
Q L Q K
=
+
dx
L x K x
= L1 K 1 x1
Mathematics for Economics - ECON 205
Final Exam
Duration: 2 hours
Assistant Professor DO Quoc-Anh, Singapore Management University
November, 2009
Introduction
This nal examination evaluates your capac
NBER WORKING PAPER SERIES
MONITORING WORKS:
GETTING TEACHERS TO COME TO SCHOOL
Esther Duflo
Rema Hanna
Working Paper 11880
http:/www.nber.org/papers/w11880
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Ma
Lecture 8
1. Su cient Condition for Local Optimum: Many Variables
2. Su cient Condition for Global Optimum: One Variable
3. Su cient Condition for Global Optimum: Many Variables
Lecture 8
(Singapore M
Lecture 10
1. Su cient Condition: Global Optimum
2. Optimization with n Variables and m Constraints
3. Envelope Theorem without constraint
4. Envelope Theorem with constraint
Lecture 10
(Singapore Man
Lecture 11
1. Optimization with Inequality Constraint: Kuhn-Tucker Conditions
2. Su ciency of Kuhn-Tucker Conditions
Lecture 11
(Singapore Management University)
Optimization V
2010
1 / 36
Review of E
Intermediate Mathematics for Economics
Final Exam
Instructions
1. This is a closed book exam.
2. No calculators or mobile phones are permitted to use during the exam.
3. You are bound by the SMU Exami
1. (10 points) If a function f is continuous on a compact set, then the function f attains
both a maximum and a minimum on the set. For example, consider
max f (x) = x s.t. x [1, 1].
x
The theorem pro
Here are selected questions from examinations from the past few years (some have been modified
slightly). More past exam questions have appeared in your recent assignments.
AT
Question 1 (differentiat
Lecture 7
1. What is Optimization?
2. Existence of Optimum: The Extreme-Value Theorem
3. Necessary Condition: One variable
4. Necessary Condition: Many Variables
5. Su cient Condition: One Variable
Le
Lecture 1
Dierentiation (Chapter 6)
Dierentiability
Rules for Dierentiation
Relationship between Shape of Function and Derivatives
Continuity
Intermediate Value Theorem
Level Curve
1
1. Derivat
Intermediate Mathematics for Economics
Solution to Sample Questions, 2010
1. A rm maximizes its prot, given xed prices p and w:
max = pf (L, a) wL,
L
where f (L, a) = La and a > 0 is constant.
a. Prov
Mathematics for Economics
Solution to Homework 1
Do the following problems in the textbook.
Chapter 6.10. #2 and #6 in page 206
Chapter 6.11. #2 and #8 in page 212.
Answers for #9 are in the textbo
Functions of One Variable
[SHS] Chapter 4
1
Intervals
A set is a collection of objects. Each object in the set is called
an element of the set.
An interval is a set of (real) numbers between, and poss
Students Manual
Essential Mathematics for
Economic Analysis
rd
3 edition
Knut Sydster
Arne Strm
Peter Hammond
For further supporting resources please visit:
www.pearsoned.co.uk/sydsaeter
Preface
This
Mathematics for Economics
Solution 2
Textbook questions:
Section 7.2 #2 and #4 in page 222-223:
Section 12.2 #6 and #8 in Page 421
Section 12.3 #4 in page 426
1
1. Use the chain rule:
dZ
Q L Q K
=
+
d
Mathematics for Economics
Solution to Homework 11
Section 14.8:
#2. Form the Lagrangian:
L = x2 + 2y 2 x (x2 + y 2 1)
a. FOC are
L01 = 2x 1 2x = 0
L02 = 4y 2y = 0
0, x2 + y 2 1, (x2 + y 2 1) = 0
b. C
Solution to Homework 12
Mathematics for Economics
1. The general solution of x + 1 x =
2
1
4
is
t
1
x = Ce 2 + .
2
The equilibrium state of the equation is
1=C+
y
1
2
and it is stable. When x(0) = 1,
Mathematics for Economics
Solution to Homework 10
Section 14.7 #1. Assume 0 a <
m
p
in a consumers problem:
max x + a ln y s.t. px + qy = m
x,y
1. a. Find the solution (x , y ).
L = x + a ln y (px + q
Mathematics for Economics
Solution 8
13.2 #6
#6. The prot function of a rm is (x, y) = px + qy x2 y 2 where p and q are prices
and x2 + y 2 are costs when the rm produces quantities, x and y.
a. First
Mathematics for Economics
Solution to homework 9
Section 14.1 #4
#4 Find x and y in the following problems:
a. min x2 + y 2 s.t. x + 2y = 4 :
L = x2 + y 2 (x + 2y 4)
L01 = 2x = 0
L02 = 2y 2 = 0
x + 2y
Mathematics for Ecoomics
Solution to Homework 5
Section 12.11. #3
Section 12.11. #4 and #6
Section 15.8. #2, #6 and #8 in page 584-585
1
1. In the equilibrium,
q1 = f (q2 , c1 )
q2 = g(q1 , c2 )
a. Th