Dynamic Programming
We have now studied two ways of solving dynamic optimization problems, one based
on the Kuhn-Tucker theorem and the other based on the maximum principle. These two
methods both lead us to the same sets of optimality conditions; they di
Problem Set 1
Due Date: October 5, 2015
Problem 1: Reduce the following matrices to row echelon and reduced row echelon
forms:
Solution:
1 1
1 0
(a)
, and
.
0 1
0 1
The first result obtained by the row operation R2 2R1 + R2 . Then the second result is
ob
Sample Problems
Most of the exam questions will be similar to the homework problems. As you can
imagine most of the questions will about matrices and vectors. You should know the
following concepts (the list may not be complete):
1. Sets, elementary set o
Problem Set 8
Due Date: December 7, 2015
Problem 1: Determine the definiteness of the following constrained quadratics.
+ 2x1 x2 x22subject to x1 + x2 = 0.
(a) Q(x1 , x2 ) = x21
0 1 1
Solution: A = 1 1 1 .
1 1 -1
Because |A| = 2, the quadratic form is neg
Suggested Solutions for Problem Set 1
Problem 1: Write the following in set notation:
(a) The set of all real numbers greater than 30.
Solution: cfw_x R|x > 30. It is OK if you used the sign.
(b) The set of all real numbers greater than 8 but less than 90
Problem Set 6
Due Date: November 9, 2015
Problem 1: Using the implicit and explicit form to compute y 0 (x) for (x, y) = (1, 1) in
the following equation: G(x, y) = y 2 5xy + 4x2 = 0.
Problem 2: Consider the function F (x1 , x2 , y) = x21 x22 + y 3 .
(a)
Midterm Exam: Econ 3700
This exam lasts one and half hour. The exam is closed book, although you are allowed
to use a basic calculator. You are expected to show all of your work. Good luck!
Problem 1. (25 points) Consider the following system of linear eq
Problem Set 5
Due Date: November 2, 2015
Problem 1: Sketch the level sets for each of the following functions from R2 R1 .
(a) f (x, y) = x2 y 2
(b) f (x, y) = y x2
(c) f (x, y) = ye2
(d) f (x, y) = x2 y 2
(e) f (x, y) = y/z
Problem 2: Compute the partial
Midterm Exam: Econ 3700
This exam lasts one and half hour. The exam is closed book, although you are allowed
to use a basic calculator. You are expected to show all of your work. Good luck!
Problem 1. (25 points) Consider the following system of linear eq
Problem Set 1
Due Date: October 5, 2015
Problem 1:
forms:
1
(a)
-2
Reduce the following matrices to row echelon and reduced row echelon
-1 -1
1
1 3 4
,
(b)
,
(c) 2 1 .
-1
2 5 7
1 0
Problem 2: Use Gauss-Jordan elimination to solve the following four system
Problem Set 7
Due Date: November 23, 2015
Problem 1: Write the following quadratic forms in matrix form.
(a) x21 2x1 x2 + x22
(b) 5x21 10x1 x2 x22
(c) x21 + 2x22 + 3x23 + 4x1 x2 6x1 x3 + 8x2 x3
Problem 2: Determine the definiteness of the following symmet
Midterm Exam: Econ 3700
This exam lasts one and half hour. The exam is closed book, although you are allowed
to use a basic calculator. You are expected to show all of your work. Good luck!
Problem 1. (25 points) Consider the following system of linear eq
Final Exam: ECON 3700
This exam lasts two hours. The exam is closed book, although you are allowed to use a
basic calculator. You are expected to show all of your work. Good luck!
Problem 1. (25 points) Consider the following function of three variables:
Problem Set 5
Due Date: November 2, 2015
Problem 1: Sketch the level sets for each of the following functions from R2 R1 .
(a) f (x, y) = x2 y 2
b) f (x, y) = y x2
(c) f (x, y) = yex
2
2
(d) f (x, y) = x y
(e) f (x, y) = y/x
Solution: This problem asks to
TheKuhnTuckerandEnvelopeTheorems
TheKuhnTuckerandenvelopetheoremscanbeusedtocharacterizethesolutionto
awiderangeofconstrainedoptimizationproblems:staticordynamic,andunderperfect
foresightorfeaturingrandomnessanduncertainty.Inaddition,thesesametworesults
p
The Maximum Principle
Here, we will explore the connections between two ways of solving dynamic optimization
problems, that is, problems that involve optimization over time. The _rst solution method is
just a straightforward application of the Kuhn-Tucker
The Necessity of the Transversality Condition at
In_nity: A (Very) Special Case
Consider a discrete-time, in_nite horizon model that characterizes the optimal consumption
of an exhaustible resource (like oil or coal). Let time periods be indexed by t =
0;
ECON 3700
Mathematics for Economists
Fall Term 2014
Instructor:
Telephone:
Oce Hours:
John Rumsey
902-494-2026 (Econ)
Mondays & Wednesdays, 12:30 - 1:30
Oce:
E-mail:
C11 6220 University Ave
[email protected]
Class Notes:
Lecture notes with solved problem