Class reps for Econ 255
Need 2 class representatives to
organize
the USAT evaluations
make the occasional class announcement
about events in the department
Solution of Linear-Equation System
We can now use the concept of inverse matrix to
solve systems
Midterm
Thursday, Feb 13, regular class time
Location:
Last names A to I:
Last names J to O:
Last names P to Z:
BOTTERELL 139
BOTTERELL 143
BOTTERELL 147
Everything that was covered in class
No books, notes, etc.
Bring student ID
Extra office hours
Extra
Important Dates
Midterm tutorial:
Monday,
Feb. 10
5:30 7:00
MacDonald 1
Assignment 4:
Due
Thursday, Feb 6, 1:55pm
Plan for Today
Rules of differentiation
Higher order derivatives
Inverse function and its derivative
Chapter 7: RULES OF
DIFFERENTIATION
Plan for Today
Matrix transpose
Inverse of a matrix and its properties
Linear dependence of equations and vectors
Matrix Transpose
If in matrix A we interchange rows and columns, we
get the transpose of A
Denoted A
Ex 1:
Note: Every matrix has a transpose
Final exam date
Final exam: April 23, 9:00am
It is the students responsibility to identify a
legitimate conflict, which is 3 consecutive exams
Ex:
9 am, 2 pm, 7 pm
or 7 pm, 9 am, 2 pm
You must report it to the exams office by March 7
Econ Department will
Textbook
Recall: Textbook is required
Bookstore will return unsold copies to
the publisher in early March
Plan for Today
How to compute A-1
Using A-1 to solve Ax = d
Cramers rule
Examples
Matrix Inversion
Matrix Inversion
Suppose A below is non-singular
Plan for Today
Concept of total derivative
Linear approximations
Differential of a function
Total derivative
Chapter 8: Comparative
Statics of General
Function Models
Example
Ex:
Y C I 0 G0
C C (Y , T0 )
=>
Y C (Y , T0 ) I 0 G0
Y * Y * ( I 0 , G0 , T0 )
Plan for Today
Rules of differentials
Total derivative
Implicit functions
Differentials
We want to know how much y changes if x changes
to x + dx
y f ( x dx) f ( x)
Approximation: The differential of y = f(x)
Denoted dy (or df):
dy f ' ( x)dx
Rules of Dif
Plan for Today
Limits
Continuity of a function
Derivative of a function
Limit
In general lim y can be different from lim y
If lim y lim y L , then the limit of y exists
We then write
x t
x t
x t
x t
lim y L
x t
If lim y or lim y , then y has no limit
Plan for Today
Implicit functions with multiple parameters
Implicit functions given by a system of equations
Higher order approximations
Implicit Functions
Implicit relationship between x and y: F(x,y) = 0
Ex:
Questions:
x2 + y2 4 = 0
1)
Can we think of
Plan for Today
Convexity and concavity in optimization
The
role of the 2nd derivative
The N-th derivative test
Multiple variables
Quadratic forms
Concavity and
the 2nd derivative
2nd and Higher Derivatives
If f(x) exists and is continuous
=> f is twice c
Chapter 9:
OPTIMIZATION
Plan
Optimization: a single endogenous variable
Stationary
1st
points
derivative test
Convexity
The
and concavity in optimization
role of the 2nd derivative
Illustration
y
x
Stationary points
Proposition 1: Suppose f is continu
Plan for Today
Implicit functions given by a system of
equations: Example
More on approximations:
Quadratic
approximations
Higher order approximations
Taylor series?
Example 2
The system below is satisfied at point A = (x,y,z,u,v,w)
= (-1,0,1,1,1,0). F
Plan for Today
Taylor series
Lagrange remainder
Optimization
The Taylor Approximation
The n-th order Taylor Approximation for f at x = a :
f ' (a )
f " (a )
f ( x) f (a )
(x a)
(x a)2
1!
2!
(n)
f (a )
.
(x a)n
n!
Example
Find the 3rd order Taylor appr
Plan
More on multivariable optimization:
Definiteness of a quadratic form
Determinantal test & examples
Characteristic roots test
The FOC
Proposition 1. If a differentiable function f(x1,.,xn)
has an interior maximum or a minimum at x* X,
then
fi(x*) = 0
Chapter 11:
MULTIVARIABLE
OPTIMIZATION
Plan
Optimization with multiple variables
Quadratic forms
Definiteness of a quadratic form Q
Definiteness of Q and concavity
Optimization & the differential
Recall: z = f(x) => dz = f (x)dx
=> FOC f (x) = 0
Similarly
Plan for Today
Matrix multiplication
Matrices and vectors
Special matrices:
Identity matrix
Null matrix
Idempotent matrix
Commutative, Associative, and Distributive Laws
Inner Product of Vectors
Suppose 2 vectors (n-tuples): x,yRn
Inner product of x and y
Plan for Today
Finish review of basic concepts
Meaning of equilibrium
Systems of equations
Matrices and vectors
Functions
Def: A function from a set X into a set Y is a rule f
which assigns to every xX a single member yY
We write y = f(x)
Also: We say f i
Page 1 of 4
QUEENS UNIVERSITY AT KINGSTON
FACULTY OF ARTS AND SCIENCE
DEPARTMENT OF ECONOMICS
ECONOMICS 255A
FINAL EXAMINATION
April 22, 2010
Instructor: M. Pak
Time Limit: 3 Hours
Instructions: Please answer all NINE questions.
Marking Scheme: Each quest
Page 1 of 12
Faculty of Arts and Sciences
Department of Economics
ECON 255
Final Examination
December 2010
Prof. James Bergin
This examination is THREE HOURS in length. There
INSTRUCTIONS; are twelve pages in this exam: the cover page and ve
questi
Page 1 of 16
HAND IN
answers on
question paper
ECON 255 - Introduction to Mathematical Economics
WINTER TERM 2011
Queens University
Department of Economics
Professor J an Zabojnik
FINAL EXAM
April 23, 2011
NAME: ID#:
SIGNATURE:
INSTRUCTIONS: