THE AUSTRALIAN NATIONAL UNIVERSITY 
Semester Two, 2022: Final Examination (Group 1) – 3 November 2022 
Mathematical Techniques for Economic Analysis 
EMET 7001 
Reading Time: Fifteen Minutes. 
Writing Time: Three Hours. 
Scanning and Submission Time: Thirty Minutes. 
Permitted Materials: This is an “open book” exam. There are no restrictions on the 
materials that may be used while undertaking this exam.. 
Page 1 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
Instructions 
1. This exam is worth sixty percent of your raw overall mark for this course. 
2. The exam consists of four questions, each of which is worth twenty-five marks. As 
such, there are a total of one-hundred marks available on this exam. 
3. An indication of the allocation of marks within each question is provided within the 
exam. 
4. Note that, in general, most of the marks for any part of any question will 
be allocated to the quality, relevance, and accuracy of the supporting 
explanation and analysis that you provide in your answer, rather than 
to the accuracy of the precise answer itself. (One implication of this is 
that a correct final answer that is accompanied by insucient supporting 
material might receive a very low mark.) This will be the case regardless 
of whether or not it is explicitly indicated within the question itself. 
5. You have three hours and forty-five minutes in which to complete this exam and 
submit it, unless the University has approved an alternative arrangement. This 
consists of fifteen minutes of reading time, three hours of writing time, and thirty 
minutes of “document scanning and submission” time. Please note that writing 
is not permitted during both the reading time and the “document scanning and 
submission” time. Writing is only permitted during the writing time. 
6. Your answers must be handwritten. They must not be typed. You are free to use 
either “paper and pen (or pencil)”, “tablet and stylus”, or any other technology 
that allows you to handwrite your solutions. 
7. Please attempt as many questions as you can in the allotted time for this exam. 
8. Please start your answer to each question on a new page and indicate which question 
is being answered at the top of the page. Where relevant, please indicate the part 
of a question that is being answered within your answer to each question. 
9. Please sort all of your answers into the same order as the corresponding questions 
appear on the exam question paper. Where relevant, please sort all of your answers 
to parts of a question into the same order as the corresponding parts appear within 
that question on the exam question paper. 
10. Please submit a digital file containing a copy of your answers to the exam before the 
expiration of the allotted three hours and forty-five minutes for this exam. The exam 
must be submitted no later than 12:45 pm on Thursday 3 November 2022 (Canberra, 
ACT, Australia Time) unless an alternative arrangement has been approved by the 
University. No late exams will be accepted. This applies even if the exam answers 
are submitted only a tiny bit late. 
11. Please submit your answers to the exam by using either the Turnitin link provided 
on the Wattle site for this course or, if you are unwilling or unable to use Turnitin, 
Page 2 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
by email to EMET7001@anu.edu.au. Any exam that is not submitted using at least 
one of these two methods might well be missed and, as a result, be considered not 
to have been submitted. In particular, please do not submit your exam to any email 
address other that the one provided above. 
12. Please note that if you are unwilling or unable to use Turnitin, you must provide 
copies of all reference material that you use during the exam if you are requested to 
do so before the release of final results for this course. 
13. Please note that this exam will be invigilated over Zoom. You must enter the Zoom 
meeting for this exam before accessing the exam question paper. You must remain 
in the Zoom meeting for this exam from the point at which you enter it until some 
point in time after you have submitted your exam answers. You must be visible on 
a web-camera for the entire time that you are in the Zoom meeting for the exam. 
No exam answers that are submitted after you leave the Zoom meeting for the exam 
will be accepted. 
14. A link for the Zoom meeting for this exam will be provided in the “Assessment 
Items 3: Final Exam” block on the Wattle site for this course. The exam Zoom 
meeting will open no later than 8:55 am on the day of the exam, in order to allow 
you to join it before the start of the exam. This is five minutes before the exam 
questions paper will become available (at 9:00 am on the day of the exam). 
15. The exam questions paper and the Turnitin submission link for the exam will also 
be provided in the “Assessment Items 3: Final Exam” block on the Wattle site for 
this course. 
16. The timing for the components of the exam is as follows. The exam Zoom meeting 
will open no later that 8:55 am on the day of the exam. The exam questions paper 
will become available, and reading time will commence, at 9:00 am on the day of the 
exam. Reading time will end and writing time will commence at 9:15 am on the day 
of the exam. Writing time will end and “document scanning and submission” time 
will begin at 12:15 pm on the day of the exam. Document scanning and submission 
time, and the exam Zoom meeting, will not end before the earlier of 12:45 pm on 
the day of the exam and the time at which the last of the students that attempt 
the exam leaves the exam Zoom meeting. Document scanning and submission time 
will end no later than 12:45 pm on the day of the exam. Hopefully, the exam Zoom 
meeting will end no later than 12:50 pm on the day of the exam. 
17. Good luck! 
Page 3 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
Question 1 (25 marks) 
1. (5 marks). Use the “definition of negative semi-definiteness of a quadratic form” to 
determine the set of values of c for which the matrix 
A = 
0@ c 0 
0 c 
1A 
is negative semi-definite. You may assume that c 2 R and that all relevant variables 
are real-valued. 
2. (5 marks). Use the “eigenvalue technique” to determine the definiteness of the 
matrix1 
B = 
0@ 3 4 
4 8 
1A . 
3. (5 marks). Use the “eigenvalue technique” to determine the definiteness of the 
matrix 
C = 
0@ 3 1 
7 5 
1A . 
4. (5 marks). Use the “leading principal minors technique” to determine the definite- 
ness of the matrix 
D = 
0BBBB@ 
1 4 1 
0 6 0 
1 2 1 
1CCCCA . 
5. (5 marks). Use the “leading principal minors technique” to determine the definite- 
ness of the matrix 
E = 
0BBBBBBBB@ 
7 0 3 0 
0 2 0 5 
3 0 4 0 
0 5 0 15 
1CCCCCCCCA 
. 
1A square matrix can be positive definite, positive semi-definite, negative semi-definite, negative defi- 
nite, or indefinite. 
Page 4 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
Question 2 (25 marks) 
1. (8 marks.) Consider a simple IS-LM model of a closed economy with no government 
sector, in which the price level is fixed. The endogenous variables in such a model 
are the equilibrium goods output (and income), which we will denote by Y , and the 
equilibrium real (an, given the fixed price level, nominal) interest rate, which we 
will denote by r. Suppose that the locus of goods market equilibrium combinations 
(that is, the IS curve) for this economy is described by the equation 
0.3Y + 100r = 252, 
and the locus of money market equilibrium combinations (that is, the LM curve) 
for this economy is described by the equation 
0.25Y 200r = 176. 
(a) Express this system of equations as a matrix equation of the form Ax = b for 
which x = (Y, r)T . 
(b) Find the cofactor for each of the elements of the coecient matrix in that 
system of equations. (Explicitly calculate each of these cofactors. Do not use 
any “short-cut” approaches to obtain them.) 
(c) Find the cofactor matrix of the coecient matrix in that system of equations. 
(d) Use the relevant cofactor matrix to find adjoint matrix of the coecient matrix 
in that system of equations. 
(e) Use the relevant adjoint matrix to obtain the inverse coecient matrix for that 
matrix equation. 
(f) Use the relevant inverse coecient matrix to find the equilibrium goods output 
and the equilibrium real interest rate for this economy. 
2. (8 marks.) Consider a simple IS-LM model of a closed economy with no government 
sector, in which private investment demand is exogenous and the price level is fixed. 
The endogenous variables in such a model are the equilibrium goods output (and 
income), which we will denote by Y , and the equilibrium real (an, given the fixed 
price level, nominal) interest rate, which we will denote by r. Suppose that the locus 
of goods market equilibrium combinations (that is, the IS curve) for this economy 
is described by the equation 
Y C (Y, r) I0 = 0, 
and the locus of money market equilibrium combinations (that is, the LM curve) 
for this economy is described by the equation 
L (Y, r) = 
M0 
P0 
. 
Note that C (Y, r) is the consumption demand function, L (Y, r) is the (real) money 
demand function, I0 is the exogenous private investment demand, M0 is the exoge- 
nous nominal money supply, and P0 is the fixed (and therefore exogenous) price 
Page 5 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
level. You should assume that 
0 < CY = 
@C 
@Y 
< 1, 
1 < Cr = @C 
@r 
< 0, 
0 < LY = 
@L 
@Y 
<1, 
1 < Lr = @L 
@r 
< 0, 
0 < P0 <1, 
and 
0 6 I0 <1. 
The impact of a small (indeed, an infinitesimally small) change in the nominal money 
supply on the equilibrium goods output ( @Y@M0 ) and the equilibrium real interest rate 
( @r@M0 ) is given by the solution to the following pair of equations:8<: 
(1 CY ) @Y@M0 Cr @r@M0 = 0, 
LY 
@Y 
@M0 
+ Lr 
@r 
@M0 
= 1P0 . 
9=; 
(a) What is the determinant of the coecient matrix in this system of equations? 
(b) Under what circumstances will this system of equations have a unique solution? 
(c) Use Cramer’s rule to find the equilibrium goods output and the equilibrium 
real interest rate for this economy. 
(d) Is it possible to determine the sign2 of the comparative static derivative @Y@M0 ? 
If so, what is its sign? 
(e) Is it possible to determine the sign of the comparative static derivative @r@M0 ? 
If so, what is its sign? 
3. (9 marks.) Consider a simple two-market linear Marshallian-cross partial competi- 
tive equilibrium economic model. The market demand function for commodity one 
is 
QD1 (P1, P2) = 18 3P1 + P2. 
The market supply function for commodity one is 
QS1 (P1, P2) = 2 + 4P1. 
The equilibrium (market-clearing) condition for commodity one is 
QD1 (P1, P2) = Q 
S 
1 (P1, P2) = Q1. 
2The sign of an object indicates whether that object is strictly negative, non-positive, zero, non- 
negative, or strictly positive. 
Page 6 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
The market demand function for commodity two is 
QD2 (P1, P2) = 12 + P1 P2. 
The market supply function for commodity two is 
QS2 (P1, P2) = 2 + 3P2. 
The equilibrium (market-clearing) condition for commodity two is 
QD2 (P1, P2) = Q 
S 
2 (P1, P2) = Q2. 
(a) Use the equilibrium condition for commodity one and the equilibrium condition 
for commodity two to express this model as a system of four simultaneous linear 
equations in four unknown variables. 
(b) Express that system of four simultaneous linear equations in four unknown 
variables as a single augmented-row matrix. 
(c) Apply the process of Gauss-Jordan elimination to that augmented-row matrix 
to obtain its reduced row-echelon form. 
(d) What is the equilibrium quantity traded of commodity one, the equilibrium 
price of commodity one, the equilibrium quantity traded of commodity two, 
and the equilibrium price of commodity two? 
Page 7 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
Question 3 (25 marks) 
Consider a consumer whose preferences over bundles of non-negative amounts of each of 
three distinct commodities can be represented by a utility function U : R3+ ! R of the 
form 
U (q1, q2, q3) = q1 + 
p 
q2q3. 
The constant per unit price of commodity one is p1 2 (0,1), the constant per unit price 
of commodity two is p2 2 (0,1), and the constant per unit price of commodity three is 
p3 2 (0,1). The consumer has an income of y 2 (0,1). 
1. What is the consumer’s budget-constrained utility maximisation problem? (1 mark.) 
2. What is the Lagrangean function for the consumer’s budget-constrained utility max- 
imisation problem? (1 mark.) 
3. (What are the first-order conditions for the consumer’s budget-constrained utility 
maximisation problem (assuming that a strictly positive amount of each commodity 
will optimally be purchased)? (8 marks.) 
4. What are the consumer’s Marshallian demands3 for each of the commodities (as- 
suming that a strictly positive amount of each commodity is optimally purchased)? 
In the case of each commodity, identify whether the Marshallian demand mapping 
is a function or a correspondence. (You may assume that appropriate second-order 
conditions for a maximum and constraint qualification conditions are satisfied.) (12 
marks.) 
5. What is the consumer’s indirect utility function (assuming that a strictly positive 
amount of each commodity is optimally purchased)? (3 marks.) 
3Marshallian demands are also known as Walrasian demands, ordinary demands, and uncompensated 
demands. 
Page 8 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001) 
Question 4 (25 marks) 
1. Find the pentic (fifth degree polynomial) Maclaurin series approximation for the 
function f (x) = 
p 
1 x. (Hint: A Maclaurin series is a Taylor series that is centred 
on the point x = 0.) (3 marks.) 
2. Is the sequence {xn}n2N, where xn = n [1 + (1)n], convergent? Prove the validity 
of your claim. If the sequence is convergent, find its limit. (3 marks.) 
3. Is the sequence {xn}n2N, where xn = [n2 + (1)n 2n]1, convergent? Prove the 
validity of your claim. If the sequence is convergent, find its limit. (3 marks.) 
4. Find limx!1 5x 
811x7+6x6+x2x 
(x1)3 . (3 marks.) 
5. Find limx!+1 
p 
1+x3 
ln(x) . (3 marks.) 
6. Use the principle of mathematical induction to prove that an = 0 for all n 2 N, 
where an = limx!+1 
[ln(x)]n 
x . (5 marks.) 
7. Use an "– argument to prove that limx!1 2x 
46x3+x2+3 
x1 = 8. (5 marks.) 
——— End of Examination ——— 
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Page 9 of 9 – Mathematical Techniques for Economic Analysis (EMET 7001)