ECON 831 - Mathematical Economics ANSWERS MIDTERM 1. (Budget correspondence)
(see your notes)
2. (Metric)
The function d is a non-negative real valued function and we show that it is a metric. We have to check the 3 properties: (i) We show that x, y Rn ,
SIMON FRASER UNIVERSITY DEPARTMENT OF ECONOMICS
COURSE: Econ 831-4 TITLE: Mathematical Economics PREREQUISITES: ECON 331 Description: The course provides a fairly rigorous treatment of the mathematical methods used in graduate economic courses. Topics tha
List of Theorems for Midterm 2009:
1) 2) 3) 4) 5)
Sequential characterization of closed sets. Any compact subset of a metric space X is closed and bounded. Blackwells contraction lemma Budget correspondence is lower hemicontinuous Theorem of Maximum, part
ECON 831 - Mathematical Economics MIDTERM
B. Antoine Time: 1h50 Fall 2008 Oct. 20th, 2008
1. [5] We dene the diameter of any bounded subset S of a metric space (X, d) as:
diam(S ) = supcfw_d(x, y ) : x, y S .
(a) Consider the metric space (R, d1 ) (recall
ECON 831 - Mathematical Economics MIDTERM Fall 2007 B. Antoine Time: 1h50 1. [5] Consider the set A = cfw_1/k, k N = cfw_1, 1/2, 1/3, .. (a) Show that A is not open in R. (b) Show that A is not closed in R. (c) Intuitively, what is the closure of A and wh
ECON 831 Homework #5
due in class Nov. 17th
Exercise 1:
Prove the following statement: If F is an algebra, then (2'): A, B F (A B ) F . A collection F of subsets of a non-empty set is an algebra if it satises: (1) A F Ac F and (2').
Exercise 2:
Prove the
ECON 831 Homework #4
due in class Nov. 5th
Exercise 1: (Consumption-savings problem)
Consider the consumption-savings problem introduced in class on Oct. 27th. Finish the resolution by backward induction. Note: assume that the problem has been written wit
ECON 831 Homework #2
due in class Oct. 15th
1. Consider the self-correspondence on [0, 1] dened as follows:
(x) = (0, 1] if x = 0 (0, x) if 0 < x 1
(i) Represent graphically. (ii) Is upper-hemicontinuous? You need to study what happens at each point x [0,
ECON 831 Homework #2
due in class Oct. 1st
Provide detailed calculations and justications to get full credit. Partial credit may be given.
Exercise 1:
Write the following mathematical statements with quantiers and implication arrows. (i) Denition of an op
ECON 831 Homework #1
due in class Sept. 24th
Provide detailed calculations and justications to get full credit. Partial credit may be given.
Exercise 1: (Human capital accumulation)
Consider a worker who enters the labor market at time t = 1 with human ca
ECON 831 Homework #1
due in class Sept. 24th
1. Consider the following metric space (X, d). Show that: |d(x, y )d(y, z )| d(x, z )
for any x, y, z X .
2. Consider the following two metric spaces with the same reference set, X , but
dierent metrics: (X, d1
Extensions of the Cake-Eating Problem
We consider the basic framework of the cake-eating problem (as stated in the midterm). The two examples below provide some further elements to this basic structure.
1. Utility in period t is now given by u(ct , ct1 ).
ECON 831 Solution Midterm 1. (a) Consider the following sequence of subsets Sk in the metric space (R, d1 ):
1 1 Sk = (1 k+1 , 1 + k+1 ). The following conditions are satised: (i) For any k , consider the real number 1. It is such that, for any k < , 1 Sk
ECON 831 Answers Homework #5 Exercise 1:
(i) We prove: If F is an algebra, then (2'): A, B F (A B ) F . Suppose F is an algebra. Pick any A and B F . Then we have:
A, B F
Ac , B c F by property (1) (Ac B c ) F by property (2) (Ac B c )c F (A B ) F
(ii)
ECON 831 Solution Homework #4 1. Bierens, 11
Consider F an algebra. (i) First, we show that F -algebra F monotone class. By denition of a -algebra, we know that for any sequence of sets in F , their innite union is also an element of F , and their innite
ECON 831 Answers Homework #4 Exercise 1: (Consumption-savings problem)
- First, we consider the 1-period problem starting at T for any state w:
VT (w) = max
c[0,w]
c
One has to maximize a strictly increasing function over a compact set. We easily deduce
Exercise 2: (Upper hemicontinuity)
(i) Consider 1 : R R, 1 (x) = (0, x2 + 1] x R. We want to show that 1 is uhc. Consider any real number x R and then 1 (x) = (0, x2 + 1]. Consider any open set O R such that (0, x2 + 1] O. Necessarily, there exists > 0 s.
ECON 831 Solution Homework #1 1. (i)
(ii) - Study at 0: (0) = (0, 1]. Preliminary remarks: NB1: there are only 2 open sets O in [0, 1] s.t. (0, 1] O, (1) [0, 1] (which is open and closed); (2) (0, 1]. NB2: for any > 0, N ,[0,1] (0) = cfw_y [0, 1] / d(0, y
ECON 831 Solutions Homework #2 Exercise 1:
(i) Consider S X : S open in X x S, > 0 / N ,X (x) S . (ii) Consider (xm ) X : xm x, x X > 0, M R / d(xm , x) < M.
m
Exercise 2:
We consider 2 equivalent metrics d1 and d2 on X . (i) We start with ": Consider a s
ECON 831 Solution Homework #1 1. Let (X, d) be a metric space, and x, y, z X . By the triangle inequality, we
have:
d(x, y ) d(x, z ) + d(y, z ) d(x, y ) d(y, z ) d(x, z ) d(y, z ) d(x, z ) + d(x, y ) d(y, z ) d(x, y ) d(x, z )
From (1) and (2), we conclu
ECON 831 Solutions Homework #1 Exercise 1:
i) The present value of her lifetime earnings is:
T T
t ht kt =
t=1 t=1
t
kt+1 kt
kt
ii) - The state variable is: kt , the current human capital stock. - The control variable is: ht the number of hours worked.
Fall 2009 ~ ECON 831 Course Material
General information:
- outline - Office#1660 - Email - Office hours: Tuesday 3 to 4pm; Thursday 2 to 3pm
Overall schedule:
(1) Dynamic optimization: * 2 Reference books: Sundaram, R. "A first course in optimization the