Chapter 1
Optimization
1.1
Introduction to maximization and minimization
Definition 1 (Maximizer, minimizer). Let S Rn and f : S R. The
point x S is said to be a (global) maximizer of f (over S) if and only if
x S
f (x ) f (x) .
The point x S is said to b
ECON 225: Mathematics for Economists
Homework II
Due: March 2, 2017
1)
Let
f : R3 R
be dened as
f(x1 , x2 , x3 ) = 4x1 x21 (x2 x3 )2 .
(x1 , x2 , x3 ) R
a. Find the gradient of
b. Does
f
f.
What is the maximum value that
2)
R3 ?
have a (global) maximizer
Name, Last name:
, ID:
, Department:
Bilkent University
ECON 225 Mathematics for Economists
Quiz 0
January 7, 2017
For the next 2 questions let f : (0, 1) R be defined as f (x) = 1/x (which is graphed below)
f
x
1
1. Which of the following statements are
ECON 225: Mathematics for Economists
Homework IV
Due: March 30, 2017
1) Is the function f : R3 R dened as
(x1 , x2 , x3 ) R3
f(x1 , x2 , x3 ) = 2x1 x2 + 2x21 + x22 + 2x23 5x1 x3
concave, convex, or neither?
2) Let f and g be two real valued convex functio
1) Let r be a positive real number and m a real number. Solve the maximization problem:
Maximize: 1 rx2 y2
Subject to: x + y = m .
For any r (0, ) and m R dene F(r, m) as
F(r, m) = maxcfw_1 rx2 y2 : x + y = m .
Find F(r, m)/r and F(r, m)/m.
2) Solve the m
ECON 225: Mathematics for Economists
Homework VI
Due: April 13, 2017
1) Let f : R2 R2 be dened as: for any x R2
x1
x
f
= 2 .
x2
x1
Show that f is a linear function. Find the matrix that represents f. Show your work.
2) Let f : R3 R2 be dened as: for any
ECON 225: Mathematics for Economists
Homework III
Due: March 9, 2017
1) A rm uses labor and capital to produce a single output. Its production function is
given by:
(l, k) R2+
f(l, k) = l1/2 k1/4 ,
where l is the hours of labor employed and k is the units
ECON 225: Mathematics for Economists
Homework VII
Due: May 4, 2017
1) A is a 4 4 matrix that can be expressed as the product of elementary matrices as
follows:
A = E2,1 (1)E2,3 E4,1 (1)E1 (3)E2,3 (4) .
a. Find A.
b. Is A invertible? If it is nd the invers
ECON 225: Mathematics for Economists
Homework V
Due: April 6, 2017
1)
Let
S Rn
be a convex set.
quasiconcave and
g:RR
Prove or give a counter example: If
is an increasing function, then
2) Prove or give a counter example: If
gf
f : S R
is
is quasiconcave.
ECON 225:
Mathematics for Economists
Homework I
Due: February 23, 2017
1) Consider the 4-dimensional Euclidean vector space R4 .
cfw_(1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1), (1, 0, 0, 1).
Let S =
a. Is the set S linearly independent?
b. Is the vector (2,
IE 325: Stochastic Models
Homework Assignment 3
Fall 2016
Due: November 21, 17:00
Solution of Question 1. (a) Let Xn be the type of exam given by the professor at time
n, the type of the exam is only dependent to the previous step, therefore it is Markov
ECON 207
Fall 2016
Exercise Set 5
1. Consider a rm which produces according to the following production
function by using labor and capital:
1
1
f (l; k) = k 4 l 2
(a) Solve the cost minimization problem of this rm for the given wage
rate, w and the renta
Exercise 7.2
Answer: A decomposition cfw_R1, R2 is a lossless-join decomposition if R1 R2 R1 or
R1 R2 R2.
Let R1 = (A, B, C), R2 = (A, D, E),
R1 R2 = A.
Since A is a candidate key (see below) , Therefore R1 R2 R1.
Starting with A BC, we can conclude: A B
IE 325: Stochastic Models
Homework Assignment 3
Fall 2016
Due: November 21, 17:00
Question 1. A professor continually gives exams to her students. She can give three possible types of exams, and her class is graded as either having done well or badly. Let
CMPUT 391 Midterm Exam (O.R. Zaane) October 24th, 2001
Page 2 of 11
ID#_
Section 1: Armstrong Axioms [10 points]
1- (6 points) State the three Armstrong Axioms (reflexivity, augmentation and
transitivity) and demonstrate (argue) that they are sound:
We kn
Math 68 - Spring 2014 - Practice problems with solutions Chapter 3
3.1.1 Find a maximum matching in each graph below. Prove that it is a maximum
matching by exhibiting an optimal solution to the dual problem (minimum vertex cover). Explain why this proves
Academic:
Carroll, Marion L., and Jay Ciaffa. "The Human Genome Project: A Scientific and Ethical
Overview." Genomics. N.p., n.d. Web.
Stamps, Julia and , Wayne, Justin. The Human Genome Project Stanford University. Web.
Garver, Kenneth L. The Human Genom
STRATEGIC AND INNOVATIVE
MARKETING
MBA 1 (A)
ASSIGNMENT
UK HOUSE PRICES
PRESENTED TO:
Ms. Ellie Semsar
Sheikh Saeed
AUGUST 2009
LONDON SCHOOL OF COMMERCE
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Exercise Set 2 (Econ 207-Kemal YILDIZ)
1 Consider your preferences for five dollar bills and ten dollar bills (and suppose
that you could have partial 10 and 5 bills).
A: Suppose that all you care about is how much money you have, but you dont
care whethe
Chapter 8 and 11
Exercises and Problems
VAT Question
During August the following transactions took place
(assume VAT is 18%)
1 Aug. Purchased merchandise for TL 28.000 on credit (Including
VAT) , VAT included
3 Aug. Received maintenance invoice for TL 1.4
8&11
Liabilities
Chapter 8
Mugan-Akman 2012
2
Liabilities
obligations of an entity to make a future payment
or to deliver goods or services to the third parties
in the future in return for cash borrowed or service
used or goods acquired
classified accor
1. NEW VS. OLD KEYNESIAN MODEL WITHIN LUCAS CRITIQUE
Common: Deman side determined, nominal rigidities (?)
Vs: *New is not all prices fixes, chance to change the price while old assumes no price change
*New is based on rational expectations theory while o
Exercise Set (covers the relevant parts of chapters 3,4,5,6,9)
1) Deniz faces a choice set X = cfw_A, B, C, D. Her preferences are defined by A
Are these preferences complete? Are these preferences transitive?
B, B
D, B
C, D
A.
2) Meryem faces a choice se
Hicksian vs Slutsky
Assume a person has a utility function U = XY, and money income of $10,000, facing an initial price of X
of $10 and price of Y of $15. If the price of X increases to $15, answer the following questions:
(a) What was the initial utility