ETF2700/ETF5970 Mathematics for Business
Assignment 2 (Semester 1, 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 30th of May 2021.
You need to ensure the following requirements:
(a) You don’t have to type your answers to each question, and a scanned copy of handwritten
answers is acceptable.
(b) If a question has sub-questions for example (1), (2), etc., please clearly indicate question
numbers.
(c) If Moodle encounters system errors after 6pm Friday the 28th of May (and before the due
time), you can email your document(s) to [email protected]. In this situa-
tion, your file name should be in the format “Surname.ID.pdf”.
(d) The lecturer and tutors will not answers 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 3 days.
(b) An assignment cannot be submitted if it is overdue by more than 3 days, due to other
arrangement during the 13th week.
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(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 the questions are asking, you need to
• study the unit’s content prior to attempting the tutorial and assignment questions. This
should enhance your ability to understand the questions.
• ask a staff member to clarify the questions 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 express-
ing 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includes material from any source, staff, students or the internet-published and unpublished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.
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Questions
There are six questions. Please attempt all questions, show all the steps of your calculations and
provide explanations to justify your answers. To obtain a full marks of a question, 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.
Question 1 (15marks)
Note: This question was excluded from Assignment 1, but it is now included in Assignment 2.
Let f(x) be a function on (0, 100), having a derivative f ′(x) and a primitive function F (x) =∫ x
0 f(t)dtdefined on the same domain. For all x ∈ (0, 100), it is known that f(x) ≥ 0 and f ′(x) ≤ 0.
(1) In Week 5, we learned that f ′(x) ≤ 0 for all x ∈ (0, 100) implies that f(x) is decreasing. Show
that this statement is true by using the properties of definite integrals. In other words, for all
a, b ∈ (0, 100), show that f(a) ≥ f(b) if a < b.
(2) We also learned that F ′′(x) = f ′(x) ≤ 0 for all x ∈ (0, 100) implies that F (x) is concave. Show
that this statement is true by using the properties of definite integrals and results from (1).
In other words, for all a, b ∈ (0, 100), show that
F
(
a+ b
2
)
≥ 1
2
(F (a) + F (b)) .
Hint: for arbitrary continuous functions h(x) and g(x) on interval [a, b],∫ b
a
h(x)dx ≥
∫ b
a
g(x)dx, if h(x) ≥ g(x) for all x ∈ [a, b].
Question 2 (15marks)
Suppose that a family’s demand for milk depends on milk price x > 0 and the family’s income
y > 0 according to the function:
z(x, y) = Ax−ayb (1)
where A, a and b are positive constants.
(a) Find a constant k (in terms of the constants a and b) such that
x
∂z(x, y)
∂x
+ y
∂z(x, y)
∂y
= kz(x, y).
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(b) Based on data for the period 1925 – 1935, it is estimated that milk demand could be repre-
sented by equation (1) with a = 1.6 and b = 2.2. Calculate the value of k in this case.
(c) For the values of a and b given in (b), assume that x = x(t) = x0 × (1.03)t and
y = y(t) = y0 × (1.06)t in (1) are both functions of time t, where x0 is the price and y0 is the
income at time t = 0. Thus, z
(
x(t), y(t)
)
= z(t) is also a function of time t. Calculate the
proportional rate of growth, which is defined as the derivative of ln (z(t)) with respect to t.
Question 3 (30marks)
A firm produces two commodities, A and B. The inverse demand functions are:
pA = 900− 2x− 2y, pB = 1400− 2x− 4y
respectively, where the firm produces and sells x units of commodity A and y units of commodity
B. Its costs are given by:
CA = 7000 + 100x+ x
2 and CB = 10000 + 6y2.
(1) Show that the firm’s total profit is given by:
pi(x, y) = −3x2 − 10y2 − 4xy + 800x+ 1400y − 17000.
(2) Assume pi(x, y) has a maximum point. Find, step by step, the production levels that max-
imize profit by solving the first-order conditions. If you need to solve any system of linear
equations, use Cramer’s rule and provide all calculation details.
(3) Due to technology constraints, the total production must be restricted to be exactly 60 units.
Find, step by step, the production levels that now maximize profits – using the Lagrange
Method. If you need to solve any system of linear equations, use Cramer’s rule and provide
all calculation details. You may assume that the optimal point exists in this case.
(4) Report the Lagrange multiplier value at the maximum point and the maximal profit value
from part (3). No explanation is needed.
(5) Using new technology, the total production can now be up to 200 units (i.e. less or equal to
200 units). Use the values from part (3) and part (4) to approximate the new maximal profit.
(6) Calculate the true new maximal profit for part (5) and compare with its approximate value
you obtained. By what percentage is the true maximal profit different from the approximate
value?
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Question 4 (15marks)
The following table contains the return on equity (in percentage) of four firms and the salary (in
thousand of dollars) of their CEOs.
Firm No. Return on Equity (%) CEO Salary ($k)
1 14.1 1095
2 22.3 937
3 16.4 1078
4 5.9 578
Denote, for firm i = 1, . . . , 4, the return on equity (in percentage) as xi and the CEO salary (in
thousand dollars) as yi. We have following observations:
(x1, y1) = (14.1, 1095), (x2, y2) = (22.3, 937), (x3, y3) = (16.4, 1078), (x4, y4) = (5.9, 578).
We wish to fit a linear function
y = m̂x+ ĉ,
with (m̂,ĉ) minimizing the sum of squared errors (SSE)
f(m̂, ĉ) = (m̂x1 + ĉ− y1)2 + (m̂x2 + ĉ− y2)2 + (m̂x3 + ĉ− y3)2 + (m̂x4 + ĉ− y4)2.
You may assume that f(m̂, ĉ) has a minimum point.
(1) The first-order conditions for stationary point(s) can be rewritten as a system of linear equa-
tions in form of
A
 m̂
ĉ
 =
 57424
3688

for some 2 × 2 coefficient matrix A to be worked out. Report the value of matrix A. No
explanation is needed.
(2) Multiply both sides of the equation above by A−1 to solve the first-order conditions derived
in part (1). Report the values of m̂ and ĉ rounded to 4 decimal places.
Question 5 (15marks)
Suppose the market price P = P (Q) can be written as a function of the market demand quantity
Q ∈ (0, 100).
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(1) Suppose the demand function is P (Q) = 400−Q and the current market demand isQ0 = 50.
Showing all steps of your working, evaluate the consumer surplus in this market
CS =
∫ Q0
0
P (Q)dQ− P0Q0
where P0 = P (Q0) is the current market price. Round off your result to 4 decimal places if
necessary.
(2) For the demand function P (Q) = 500− 12Q2, evaluate the consumer surplus
CS = CS(Q0) as a function of Q0 ∈ (0, 100). Show that CS(Q0) is strictly increasing with Q0
by checking the sign of its derivative.
(3) Show that CS(Q0) is strictly increasing for any demand function satisfying P ′(Q0) < 0 for all
Q0 ∈ (0, 100).
Question 6 (10marks)
Let’s assume there exists a path satisfying an iso curve function f(x, y), where f(x, y) = d and d
is a given value. The function f(x, y) is defined for any value of x and positive y. The following
are partial derivatives of f(x, y):
∂f(x, y)
∂x
= fx(x, y) = 4e
4x(y2 − 3y), ∂f(x, y)
∂y
= fy(x, y) = 2e
4xy.
If we assume that the implicit function y = g(x) is defined by the isoquant (that is, the iso
curve) and the derivative of g(x) exists, find the analytical form of g(x) that passes through the
point (0, 5).
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