The Viner—Wong Envelope Theorem
Eugene Silberberg
The envelope theorem. now the fundamental tool in modern duality analysis.
had its beginnings in Jacob Viner’s classic 1931 article on short» and long~run
cost curves. It seemed wrong to Viner that at any
14.102 Problem Set 3
Due Tuesday, October 18, in class
1. Lecture Notes Exercise 208: Find
numbers.
Rb
a
log(t)dt , where 0 < a < b are real
2. (Sundaram 4.4, page 110) Find and classify all critical points (local maximum, local minimum, neither) of the f
14.102 Problem Set 1 Solutions
1. Lecture Notes Exercise 12: Show that Q, the set of real rational numbers,
does not have the least upper-bound property.
Solution: To show this, we must show that there exists a set E Q such
that E is nonempty and bounded
14.102 Problem Set 1
Due Thursday, September 22, in class
1. Lecture Notes Exercise 12: Show that Q, the set of real rational numbers,
does not have the least upper-bound property.
2. Is the set of real irrational numbers countable?
3. For x R1 and y R1 ,
14.102 Problem Set 2 Solutions
1. Lecture Notes Exercise 105: Given an m n matrix A, show that S(B)
S(A) and N (A0 ) N (B 0 ) whenever B = AX for some matrix X. What
is the geometric interpretation? (Note: this is a repeat from last years
problem set; as
14.102 Problem Set 3
Due Tuesday, October 18, in class
1. Lecture Notes Exercise 208: Find
numbers.
Rb
a
log(t)dt , where 0 < a < b are real
Solution: This can be solved by integration by parts.
t, F 0 (t) = f (t) = 1, G(t) = log(t), G0 (t) = g(t) = 1t ;
14.102 Problem Set 2
Due Thursday, October 6, in class
1. Lecture Notes Exercise 105: Given an m n matrix A, show that S(B)
S(A) and N (A0 ) N (B 0 ) whenever B = AX for some matrix X. What
is the geometric interpretation? (Note: this is a repeat from la
14.102 Midterm Exam Solutions
October 20, 2005
Real Analysis (15 points)
1. (2 points) Give an example of a bounded sequence with exactly three limit points.
Solution: One example is the sequence cfw_1,2,3,1,2,3,1,2,3 , .,. which
.
has the
integers 1,2, a
Solutions to Review Questions for 14.102 Midterm
10/14/05
Note: For true/false questions you should either prove the statement
or provide a counterexample.
1
Real Analysis
1. Give an example of a relation R that is transitive, but not symmetric.
Solution:
Review Questions for 14.102 Midterm
10/14/05
Note: For true/false questions you should either prove the statement
or provide a counterexample.
1
Real Analysis
1. Give an example of a relation R that is transitive, but not symmetric.
2. Suppose S is an ord
14.102 Midterm Exam
October 20, 2005
Instructions: This is a closed book exam; you have 90 minutes to complete it.
Please answer all questions.
Suggestion: Read through all questions briefly before you being, and pay attention
to the allocation of points.
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/22/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by Stavros Panageas
for the MIT Math for Economists Course i
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/13/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/4. I
have made only minor changes to the order of pres
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/8/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/4. I
have made only minor changes to the order of prese
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/27/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by Stavros Panageas
for the MIT Math for Economists Course i
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/18/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/4. I
have made only minor changes to the order of pre
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/4/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/4. I
have made only minor changes to the order of pres
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/13/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/4. I
have made only minor changes to the order of pre
14.102, Math for Economists
Fall 2005
Lecture Notes, 10/6/2005
These notes are primarily based on those written by Andrei Bremzen for
14.102 in 2002/3, and by Marek Pycia for the MIT Math Camp in 2003/4. I
have made only minor changes to the order of pres
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/20/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by Stavros Panageas
for the MIT Math for Economists Course i
14.102, Math for Economists
Fall 2005
Lecture Notes, 9/29/2005
These notes are primarily based on those written by George Marios Angeletos
for the Harvard Math Camp in 1999 and 2000, and updated by Stavros Panageas
for the MIT Math for Economists Course i