Mathematics for Graduate Students in Economics
Solutions: Exam 1
Jean Guillaume Forand
Fall 2013, Waterloo
1.a. Suppose that Archibald got his quantiers mixed up and thought the denition of a limit
was the following (wrong) one: xn x if, for any N N, ther
Mathematics for Graduate Students in Economics
Assignment 5
Jean Guillaume Forand
Fall 2013, Waterloo
1. Prove the following theorem: Fix a set C Rn and a function f : C R . Then
x arg maxxC f (x) if and only if x arg minxC f (x) and x arg minxC f (x) if
Mathematics for Graduate Students in Economics
Assignment 4
Jean Guillaume Forand
Fall 2013, Waterloo
1. If the sets A, B Rn are convex, show that the set A B is also convex.
2. Show that the following sets are convex.
(a) A = cfw_x Rn : V x = y, given n
Mathematics for Graduate Students in Economics
Assignment 3
Jean Guillaume Forand
Fall 2013, Waterloo
1. Use the chain rule to nd the derivative of the function
1
(a) f (x) = ex 2 .
(b) f (x) = log x2 .
2. Consider the function f (x) = x2 + 1.
(a) Find th
Mathematics for Graduate Students in Economics
Assignment 1
Jean Guillaume Forand
Fall 2013, Waterloo
1. Let A and B be any sets. Show that A = B if and only if A B and B A.
2. Let A and B be any sets. Use the property derived in Problem 1 to show that A
Mathematics for Graduate Students in Economics
Assignment 2
Jean Guillaume Forand
Fall 2013, Waterloo
1. Show that
(a) span
1
1
,
0
2
(b) For any a R, span
= R2 .
a
a
,
1
1
= cfw_(x, y ) R2 : x = y .
2. Show that the unit vectors cfw_e1 , e2 , e3 form a
Mathematics for Graduate Students in Economics
Assignment 6
Jean Guillaume Forand
Fall 2013, Waterloo
1. Let (, E , P ) be a probability space. Show the following properties of P
(a) First show the following result about sets: If A and B are sets, then (A
Mathematics for Graduate Students in Economics
Solutions: Assignment 1
Jean Guillaume Forand
Fall 2013, Waterloo
1. Let A and B be any sets. Show that A = B if and only if A B and B A.
Solution. We show that the condition that A B and B A is both necessar
Mathematics for Graduate Students in Economics
Solutions: Assignment 6
Jean Guillaume Forand
Fall 2013, Waterloo
1. Let (, E , P ) be a probability space. Show the following properties of P
(a) First show the following result about sets: If A and B are se
Solutions: Midterm Exam
ECON 601 Micro I
Jean Guillaume Forand
Fall 2011, Waterloo
A correct answer must include a correct explanation. Please be as precise as you can.
1. (10) Assume that the set of alternatives is X = R2 . Dene the lexicographic prefere
Mathematics for Graduate Students in Economics
Solutions: Assignment 5
Jean Guillaume Forand
Fall 2013, Waterloo
1. Prove the following theorem: Fix a set C Rn and a function f : C R . Then
x arg maxxC f (x) if and only if x arg minxC f (x) and x arg minx
Mathematics for Graduate Students in Economics
Solutions: Assignment 4
Jean Guillaume Forand
Fall 2013, Waterloo
1. If the sets A, B Rn are convex, show that the set A B is also convex.
Solution. Fix any x, y A B and any [0, 1]. Then since A and B are con
Mathematics for Graduate Students in Economics
Solutions: Assignment 2
Jean Guillaume Forand
Fall 2013, Waterloo
1. Show that
(a) span
1
1
,
0
2
= R2 .
Solution. Fix any x R2 . We need to show that we can write x as a linear combi1
0
nation of the vectors
Mathematics for Graduate Students in Economics
Solutions: Assignment 3
Jean Guillaume Forand
Fall 2013, Waterloo
1. Use the chain rule to nd the derivative of the function
1
(a) f (x) = ex 2 .
1
Solution. f (x) =
1
ex 2 .
2x
(b) f (x) = log x2 .
2
Soluti