ETF2700/ETF5970 Mathematics for Business

Assignment 1 (Semester 2, 2021)

Submission

This assignment contributes 10% to the overall assessment. You must submit all pages of

your answers on Moodle. An assignment cover sheet (with your detailed information)

is required to be attached as the front page of your submission. In case that you are

unable to merger two documents into one, you are allowed to upload the cover sheet as a

separate document.

The due time is 23:50 Sunday the 12th of September 2021.

You need to ensure the following requirements:

(a) You need to type your answer to each question in a document using Microsoft Word

(or LaTex via Overleaf).

(b) If a question has sub-questions for example (1), (2), etc., please clearly indicate

question numbers.

(c) If Moodle has system errors after 6pm on the due date, you can email your docu-

ment(s) to [email protected] In this situation, your file name should

be in the format “Surname.ID.pdf” or “Surname.ID.docx”.

(d) The lecturer and tutors will not answer any request or question that is directly related

to assignment questions before the due time.

Further Information

(a) A penalty of 10% of the total mark of this assignment will be deducted for each day

overdue, up to 4 days.

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(b) An assignment cannot be submitted if it is overdue by more than 4 days, except for

an approval from the chief examiner.

(c) Extensions beyond the due date will only be allowed in special circumstances. You

may visit

https://www.monash.edu/exams/changes/special-consideration

for the university policy and application procedure for special consideration.

(d) If you don’t understand what a question asks, you need to

• study the unit’s content prior to attempting the tutorial and assignment ques-

tions. This should enhance your ability to understand the questions.

• ask a staff member to clarify the question for you. A staff consultation schedule

is on Moodle.

Avoid Plagiarism!

Intentional plagiarism amounts to cheating. See the Monash Policy.

Plagiarism: Plagiarism means to take and use another person’s ideas and or manner

of expressing them and to pass these off as one’s own by failing to give appropriate

acknowledgement. This includes material from any source, staff, students or the internet-

published and unpublished works.

Collusion: Collusion is unauthorised collaboration with another person or persons.

Where there are reasonable grounds for believing that intentional plagiarism or collusion

has occurred, this will be reported to the Chief Examiner, who may disallow the work

concerned by prohibiting assessment or refer the matter to the Faculty Manager.

Questions

Important: There are six questions. Please attempt all the questions, show all the steps of

your calculations, and provide explanations to justify your answers. To obtain full marks,

it is important to provide complete answers supported by logically sound explanations,

unless the question explicitly states that no explanation is needed. It is not sufficient to

simply provide calculator instructions.

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Question 1 (20marks)

Suppose a company called “GlacierTech” produces glasses for smart phones and has

a market share of 25%. The market demand QD and market supply QS are both linear

functions of the market price P given by:

QD = −3P + 100, and QS = 2P − 20.

(1) Suppose that market clearing can be reached and expressed as QS = QD, and that

market demand (or supply) is a linear function of GlacierTech’s supply Q (which is

your job to specify). Hence, the price P and GlacierTech’s supply Q satisfies a system

of linear equations in matrix form given by

A

P

Q

=

100

20

,

where A is a 2× 2 coefficient matrix that you need to work out.

Specify the relation between QD (hence QS) and Q, and write down the values of

matrix A without any further explanation.

(2) Solve, step by step, the system of linear equations in the above question using

Cramer’s rule. Explain your answers carefully.

(3) In the same market, suppose the government enforce a policy on the price and

market sales volume (in terms of Q) as follows:

P = −2Q+ 90.

Do you think market-clearing can still be reached? Explain.

Question 2 (10marks)

Let

A =

1 −2

−5 12

, C =

4 −3

−1 2

, b1 =

−1

2

, b2 =

−1

5

, b3 =

1 −4

−5 3

(1) Calculate, step-by-step, the inverse matrices A−1 and C−1 .

(2) Use A−1 and C−1to solve X, Y and Z of the following linear systems (one-by-one):

AX = b1, YC = b2, AZC = b3

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Question 3 (10marks)

Solve the following linear programming problems:

max 3x+ 2y subject to

x+ y ≤ 70

x+ 3y ≤ 240

x+ 3y ≤ 90

x ≥ 0, y ≥ 0

You can assume the optimal solution exists (therefore, you don’t need to prove such an

existence). You don’t need to draw a graph to demonstrate the problem, but you need to

explain in details how to determine the corner points.

Question 4 (22marks)

Suppose that a company called A1 is the only producer, which produces professional

camera lens being attachable to smartphones, and assume that A1 can determine the

market price P ∈ (4, 10) (in hundred dollars). The market demand Q satisfies a function

in P given by

Q = −2P 2 + 15P + 36

(1) Compute the price elasticity of demand that is ElPQ(P ), at P = 5 (keep 4 decimal

places). Interpret the result from an approximation point of view.

(2) Write down the total revenue function in P , where the domain needs to be specified.

(3) Compute the derivative and the stationary points of the total revenue function de-

rived in Question 4(2).

(4) Show that using the second derivative, the total revenue function derived in Question

4(2) is concave.

(5) Solve the maximum value of the total revenue function derived in Question 4(2).

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Question 5 (16marks)

Compute, step-by-step, the first and second derivatives and the maximum value of the

following functions. You may assume that the maximum points exist in all cases.

(1) f(x) = x4 − 8x2 + 2, x ∈ [−1, 3]

(2) f(x) = 1

x2+1

, x ∈ [0, 2]

(3) f(x) = −e2x + 2ex + 2, x ∈ (−∞,∞)

(4) f(x) = − ln(x4 + 4x2 + 5), x ∈ (−∞,∞)

Question 6 (22marks)

An ice-cream lover has a total of $20 to spend one evening. The price of ice-cream is p

dollars per litre. This person’s preferences for buying q litres of ice-cream and leaving a

nonnegative amount (20− pq) dollars to spend on other items, are represented by the

following utility function:

U(q) = ln(q) + ln (20− pq), q ∈

(

0,

20

p

)

.

Note that ln(x) is the natural logarithm function in the sense that x = exp(ln(x)).

(1) Find the first-order condition for a utility, which maximises the quantity of ice-cream.

(2) Solve the first-order condition derived in Question 6(1) and express the utility, which

maximises the quantity and is denoted as q∗, as a function of p.

(3) What guarantees that your quantity q∗ is really a maximum?

(4) Express the elasticity of demand for ice-cream as a function of the price p dollars per

litre. When the price is $2.50 per litre, calculate the price elasticity of this person’s

demand for ice-cream.

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