The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 9 (no need to hand in)
1. Text book questions:
Page #
374
341
355
: no need
Exercise #
Question #
12.4
4a, 4b , 4c , 7, 8
11.6
2
12.2
1a, 1b
to check their graphs since the 3D graphs could be tough
2. Consider the function
f (x; y) = x y
2
dened
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructors Manual
CHAPTER 9
Exercise 9.2
1. (a) f 0 (x) = 4x + 8 = 0 i x = 2; the stationary value f (2) = 15 is a relative maximum.
(b) f 0 (x) = 10x + 1 = 0 i x = 1/10; f (1/10) = 1/20 is
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructors Manual
CHAPTER 8
Exercise 8.1
1.
(a) dy = 3 x2 + 1 dx
(b) dy = (14x 51) dx
(c) dy =
2. M Y
1 x2
(x2 + 1)
2
dx
dM
marginal propensity to import
= dY =
M
average propensity to impo
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Topic 6 Two variable maxmization:
Consider a twovariable differentiable function z = f (x, y) defined on S, (xo, yo) is
an interior point of S
(xo, yo) is said to be local maximum point off iff (x, y) <= f (xo , yo ) for all pairs of
(x, y) in S that l
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Topic 6 Two variable optimization
1.The Hessian Matrix
Defination: A square matrix of 2nd partial derivatives,given y=f(x1,x2,.xn)
H = f11 f12 f1n
f12 f22. f2n
.
fn1 f2n fnn
fij (i,j are both integers) stands for the second order derivative first with res
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Attention:
The Midterm exam will cover:
Topic 13 + Part of topic 4
7.2
Quotient rule:
Inclass exercises:
f(x)=2x3; g(x)=x+1
Find the derivative of f(x)/g(x)
Solution: (Use the formula directly)
f(x)=2 g(X)=1
f(x)/g(x)=[f(x)g(x)f(x)g(x)]/g(x)=[2(x+1)
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Topic 5 Total derivatives
Total differentials(for twovariable function)
z=f(x,y) ,to find the derivative of z,i.e.(df)
We take derivative with respect to x and y separately,and add them together,
df=f(dx)+f(dy)
Application:
Elasiticity and Revenue
Given
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
7.4 Partial difference
Consider a function Z=f(x,y)=x+y
If y is held constant, i.e.y=a, z=x+a dz/dx=3x
If x is held constant,i.e.x=a,z=a+y dz/dy=2y
b)
f=lim(x>0)[f(x+x,y)f(x,y)]/x
The same method applies to fy
e.g.1 A company manufacture skis
The lighte
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
8.4 Total Derivative
Recall Partial derivative f(x,y)
e.g: y=f(x1,x2,w) where x1=g(w) x2=h(w)
dy/dw=?
*w is the ultimate source of change
e.g.1 z=x^28xyy^3, where x=3t,y=1t Find dz/dt
Solution:
dz/dt=18t8(33t3t)3(1t)^2*(1)=24+66t+3(1t)^2=3t^2+6
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Topic 5:Total derivatives(continued)
Derivatives of implicit functions
F(X,Y)=0
F(y)0 Y=dy/dx=F(x)/F(y)
F(x)0 X=dx/dy=F(y)/F(x)
eg. Find the dy/dx & dz/dx in the equation set below.
x+y+z=1
(1)
x+y+z=0
(2)
Solution:
Step 1:Check when the system defines
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Topic 6 Economic applications
Example1: (problem of a multiproduct firm, P331 Example 1)
 A twoproduct firm Wlder circwnstances of pure competition, prices of two products:
p1 and p2 are given.
 Revenue of firm: R = p1Q1 + p2Q2 where Q1, Q2 are the ou
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Topic 7: Constrained optimization
In some optimization problems, the variables to be chosen are often required to
satisfy certain constraints.
For example, a conswner has a budget constraint P1X1 + P2X2 = m
Consider the 2variable utility maximization
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Final Exam Fall 2014
Dec. 18, 2014
Answer all the following questions, full work must be shown.
Calculators are not allowed.
Duration of exam: 2.5 hours
1. [30 marks, 10 marks each] Three (unrelated) questions on Concavity/convexity and quasiconcavity/con
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 8 (Assignment 3)
Due date: Wednesday, November 9
How to submit:
Method 1 Submit to your TA in person before tutorial starts.
Method 2 Submit through submission box. Deadline for Method 2: 9am
Details of submission box:
Submission Box #2 (Labeled
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 10 (no need to hand in)
1. Text book questions:
Page #
Exercise #
Question #
355
12.2
3a
: determine whether the stationary points are local maximum (minimum)
2. Use the Lagrange multiplier method to nd the stationary points of the following
prob
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 2 (Assignment 1)
Due date: Wednesday, September 21
How to submit:
Method 1 Submit to your TA in person before tutorial starts.
Method 2 Submit through submission box. Deadline for Method 2: 9am
Details of submission box:
Submission Box #2 (Labele
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 11 (Assignment 4)
Due date: Wednesday, November 30
How to submit:
Method 1 Submit to your TA in person before tutorial starts.
Method 2 Submit through submission box. Deadline for Method 2: 9am
Details of submission box:
Submission Box #2 (Labele
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 3 (No need to hand in)
1. (Question 5.6.1 on page 111 of the text book) Given the following national income
model:
8
< Y = C + I0 + G0
C = a + b (Y T )
:
T = d + tY
where a > 0; d > 0; 0 < b < 1, and 0 < t < 1 are known parameters and I0 and G0 a
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 4 (No need to hand in)
1. Consider the function f (x; y) = x2
y
(a) Draw the three level curves f (x; y) =
1; 0; 1 in the same graph
(b) Verify that (1; 2) ; ( 1; 1) ; (2; 3) are the three points on the three level curves
respectively. Find the g
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 7 (no need to hand in)
1. Questions from textbook
Page #
250
254
312
Exercise #
9.5
9.6
11.3
Question #
3a (function in Question 2a)
2
1b, 1c, 2, 4d, 4e
2. Check the following matrix for deniteness
0
2
@ 1
0
1
6
4
1
0
4 A
3
3. Determine the value
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 5 (Assignment 2)
Due date: Wednesday, October 12
How to submit:
Method 1 Submit to your TA in person before tutorial starts.
Method 2 Submit through submission box. Deadline for Method 2: 9am
Details of submission box:
Submission Box #2 (Labeled
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 1
No need to hand in
There will be tutorial starting next week
2
;)
1. Provide the graph of the following sets (except for obvious sets such as R2 ; R+
(a)
2
(x; y) 2 R+
: xy
1
2
(x; y) 2 R+
: xy
1
n
2
2
(c) (x; y) 2 R2 : e (x +y )
(b)
(d)
(e)
n
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
Exercise 6 (no need to hand in)
1. Consider the equation:
xy
zez
1=0
(1)
(a) Verify that (x; y; z) = (1; 1; 0) is a point satisfying Equations (1).
(b) Argue that Equations (1) implicitly dene z as dierentiable function of (x; y) for
(x; y; z) close to (1
The Hong Kong University of Science and Technology
Fundamental methods of mathematical economists
ECON 2174

Fall 2014
From part (b), z = f (x; y) is a dierentiable function for (x; y; z) close to (1; 1; 0)
From part (c), at (x; y; z) = (1; 1; 0):
@z
@x
Part (d), check slide 19 of Topic 3:
z
0:02, thus z(1:1; 0:99)
z(1; 1)
@z
= 0; @y
=
zx
0:02 =
2
x + zy
y = (0)(0:1) + (