ETF2700/ETF5970 Mathematics for Business project 2
Semester 2, 2018






Submission

This project comprises 10% of the assessment for ETF2700/ETF5970. You must submit a “hard copy” of your written work (with an project Cover Sheet – from the “projectS” section of Moodle) by 6pm on Monday 17 September 2018. Submit it to your tutor in your tutorial (or to your tutor’s mailbox, 5th floor H Block).

ENSURE that you


• Submit a printed “hard copy” of your project to your tutor. Your project must be typed.

• Name your project: Surname-Initials A2.docx, e.g., Einstein-A A2.docx.


• Upload this file to Moodle as follows:

Go to the “projectS” section. Click on the “Submit your project 2 here – Due  on 17 Sep” link to upload. The following message will appear momentarily, “File uploaded successfully.”

[To later confirm your upload was successful, go to the “projectS” section and click on the project 2 uploading link. The uploaded files name will be shown.]

NB, DO NOT submit any Excel files. You may upload only ONE file.


• Submit both hard copy and electronic copy before the due, otherwise your submission will NOT be accepted.

• Retain your marked project until after the publication of final results for this unit.



Further Information

• A maximum penalty of 10% of the total mark allocated to this assessment will be deducted for each day that it is late, up to 4 days. An project may not be submitted if it is late by more than 4 days.

• 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.


• Do not submit your project in a folder: stapled pages are easier for the marker.


• Save trees! Double-sided printing is encouraged.


• If you don’t understand what the questions are asking,


– study the unit’s content prior to attempting the tutorial and project questions. This should enhance your ability to understand the questions.

– ask a staff member to clarify the questions for you. A staff consultation roster 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 ex- pressing 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 This includes material from any source, staff, students or the internet-published and unpublished works.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.
Total marks: 100

[This project is worth 10% towards the final mark of this unit].


Question 1 (20 marks)


An ice-cream lover has a total of $10 to spend one evening. The price of ice-cream is $p per litre. The person’s preferences for buying q litres of ice-cream, leaving a nonnegative amount $(10 − pq) to spend on other items, are represented by the utility function:image.png

(a) Find the first-order condition for a utility maximizing quantity of ice-cream. [Hint: the square root function is a power function with exponent 1 .]

(b) Solve the first-order condition derived in (a) in order to express the utility maximizing quan- tity q∗ as a function of p.

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

(d) Express the elasticity of demand for ice-cream as a function of the price $p per litre. When the price is $2.50 per litre, calculate the price elasticity of the person’s demand for ice-cream.


Question 2 (20 marks)
Suppose that a representative Norwegian family’s demand for milk depends on the price p > 0

and the family’s income r > 0 according to the function:

image.png

where A, a and b are positive constants.


(a) Find a constant k (in terms of the constants a and b) such that





(b) Based on data for the period 1925–1935, it is estimated that Norwegian milk demand could be represented by equation (1) with a = 1.5 and b = 2.08. Calculate the value of k in this case.

(c) For the values of a and b in part (b), suppose moreover that p = p(t) = p0(1.06)t and r = r(t) = r0(1.08)t in (1) are both functions of time t where p0 is the price and r0 is the income at time t = 0. Hence, E(p(t), r(t)) = E(t) is also a function of time t. Calculate the proportional rate of growth, that is, the derivative of function ln (E(t)).


Question 3 (20 marks)

The profit obtained by a firm from producing and selling x and y units of two brands of a com- modity is given by


P (x, y) = −0.1x2 − 0.2xy − 0.2y2 + 47x + 48y − 600.

(a) Assume P (x, y) has a maximum point. Find, step