Name: _ Date: _
1. According to the rule of 70, if a country's real GDP per capita grows at an annual rate of 2%
instead of 3%, it will take _ for that country to double its level of real GDP per capita.
A) 35 additional years
B) 11.67 additional years
C)
Faculty of Arts
Department of Economics
ECN 230 Mathematics for Economics
Fall 2016 - Sections 011, 021, 031
KHE221, Wednesday 3 pm-6 pm
Lab: Thursday, KHS134 12 pm-3 pm (021, 031, 011)
Professional/Professionally Related Course
Lectures are 3 hours per w
RYERSON UNIVERSITY
DEPARTMENT OF ECONOMICS
ECN230 - Mathematics for Economics
Sample Midterm Test II - Fall 2016
Prof. C. Angyridis
November 16th, 2016
Instructions: Answer all FOUR of the following questions. Each question is worth 25 MARKS. Present your
Department of Economics
Ryerson University
ECN230-Mathematics For Economics
Problem Set #5
Please attempt the following exercises from the textbook:
1. Section 11.1: Problems 1 and 5.
2. Section 11.2: Problems 1-5, 7 and 8.
3. Section 11.5: Problems 1 and
Department of Economics
Ryerson University
ECN230-Mathematics For Economics
Problem Set #7
Please attempt the following exercises from the textbook:
1. Section 12.7: Problems 1, 2 and 5.
2. Section 12.8: Problems 1, 2, 4 and 5.
3. Section 12.9: Problems 1
Department of Economics
Ryerson University
ECN230-Mathematics For Economics
Problem Set #8
Please attempt the following exercises from the textbook:
1. Section 12.11: Problems 1-3 and 6-9.
2. Review Problems for Chapter 12: 13, 15 and 17.
These exercises
Department of Economics
Ryerson University
ECN230-Mathematics For Economics
Problem Set #6
Please attempt the following exercises from the textbook:
1. Section 12.1: Problems 1-4.
2. Section 12.2: Problems 1-5.
3. Section 12.3: Problems 1, 2, 4 and 6.
4.
RYERSON UNIVERSITY
DEPARTMENT OF ECONOMICS
ECN230 - Mathematics for Economics
Midterm Test II Sample - Winter 2016
Instructor: Ugochi Emenogu
March 11, 2016.
Name:
Student ID:
Instructions
1. Answer all FOUR of the following questions.
2. Each question is
ECN230-Mathematics for Economics
Lecture #7
1
Tools for Comparative Statics
1.1
General Homogeneous Functions
Consider the following function defined in domain :
(1 )
Assumption:
(1 ) 0 = (1 )
Then, function is homogeneous of degree if:
(1 ) = (1 )
ECN230-Mathematics for Economics
Lecture #8
1
Tools for Comparative Statics
1.1
Dierentiating Systems of Equations
Dierentials provide an ecient way to find the partial derivatives of
functions defined implicitly by a system of equations.
Example: A mac
ECN230-Mathematics for Economics
Lecture #2
1
Matrix and Vector Algebra
1.1
The Transpose
The transpose of the matrix is denoted by 0 and is obtained
by interchanging the rows and columns of .
11 1
11 1
. 0 = .
.
= .
.
.
.
1
1
In other words:
=
ECN230-Mathematics for Economics
Lecture #9
1
Multivariable Optimization
1.1
Single Variable Optimization
Consider the function:
= ()
For = 0 to be a maximum or a minimum point for in its domain,
a necessary (or first-order) condition is that it is a s
ECN230-Mathematics for Economics
Lecture #1
1
Matrix and Vector Algebra
Economists = use mathematical models these models typically
involve a system of several equations.
Linear (or Matrix) Algebra = useful in studying large systems of
linear equations.
ECN230-Mathematics for Economics
Lecture #4
1
Determinants and Inverse Matrices
1.1
Solving Systems of Linear Equations By Matrix Inversion
Suppose that:
| 6= 0
If that is the case, then:
= = = 1
1.2
Cramers Rule
A method of solving for each variable
ECN230-Mathematics for Economics
Lecture #3
1
Determinants and Inverse Matrices
1.1
Determinants of Order 3
We would like to calculate the determinant of a 3 3 matrix using the
method of expansion by cofactors:
11 12 13
| = 21 22 23
31 32 33
We hav
ECN230-Mathematics for Economics
Lecture #5
1
Functions of Many Variables
1.1
Introduction
Models in Economics involve relationships not only between two variables (say, and ), but between many variables (say, and 1 , 2
and so on).
1.2
Functions of Two V
ECN230-Mathematics for Economics
Lecture #6
1
Tools for Comparative Statics
1.1
Introduction
Economic Models:
- Variables
- Parameters
Solution Equilibrium: Variables expressed as a function of the
parameters of the model.
Comparative Statics: How does
ECN230-Mathematics for Economics Sample Final
Exam
Instructions
1. Answer all FOUR of the following questions.
2. Each question is worth 25 MARKS.
3. Present your work clearly and in detail.
4. Exam Duration: 180 minutes.
QUESTION 1
Consider the standard
ECN230-Mathematics for Economics Lecture 1
1
Matrix and Vector Algebra
Most mathematical models used by economists involve a system of several equations
such as
y1 = ax1 + bx2
y2 = cx1 + dx2
Input-output analysis is one of the areas of economics that us
Department of Economics
Ryerson University
ECN230-Mathematics For Economics
Problem Set #1
Please attempt the following exercises from the textbook:
1. Section 15.1: Problems 1, 2 and 6.
2. Section 15.2: Problems 1-4.
3. Section 15.3: Problems 1-5.
4. Sec
Ryerson University
Department of Economics
Hakan Toksoy
ECN230-Mathematics for Economics, Summer 2016
Problem Sets
Lecture VII to Lecture X
Question 1. Find the local maximum and minimum values and saddle point(s) of
f (x, y) = x4 + y 4 4xy + 1.
ANSWER:
A
Ryerson University
Department of Economics
Hakan Toksoy
ECN230-Mathematics for Economics, Summer 2016
Problem Sets
LECTURE VI
Question 1. Find dY /dt when Y (K, L) = K 2 + L3 with K(t) = t2 and L(t) = 2t by
Chain Rule.
ANSWER
dY /dt = 4t3 + 24t2
Question
Ryerson University
Department of Economics
Hakan Toksoy
ECN230-Mathematics for Economics, Summer 2016
Problem Sets-Functions with More Variables
Question 1. Consider the function f that, to every pair of numbers (x, y), assign
the number 2x + x3 y 3 . The
RYERSON UNIVERSITY
DEPARTMENT OF ECONOMICS
ECN230 - Mathematics for Economics
Midterm Test II Sample - Winter 2016
Instructor: Ugochi Emenogu
March 11, 2016.
Name:
Student ID:
Instructions
1. Answer all FOUR of the following questions.
2. Each question is
RYERSON UNIVERSITY
DEPARTMENT OF ECONOMICS
ECN230 - Mathematics for Economics
Sample Midterm Test I - Winter 2016
Instructor: Ugochi Emenogu
Feb 4, 2016.
Name:
Student ID:
Instructions
1. Answer all FOUR of the following questions.
2. Each question is wor