MATH96011 – Mathematics of Business and Economics                                                                      Spring 2021

Coursework

Hand-out date: Monday 1st March 2021, 12 noon UK time

Hand-in date: Monday 15th March 2021, 1pm UK time

Please read the following instructions carefully:

- This coursework counts for 10% of your fifinal mark in Mathematics of Business and Economics.

- Throughout this coursework, you should show any working and state clearly any additional assumptions that you make, or results from the lectures or problem sheets that you use; marks may be deducted if you fail to do so.

- Answers may be handwritten or typed, but must be presented clearly.

- Please do not use red or green ink.

- Submit a PDF file via Blackboard (see Turnitin Assignment: Coursework 1 in the folder Coursework) and name the file as follows: MATH96011 CW followed by your first and last name, e.g. if I were to submit a coursework, I would submit: MATH96011 CW Ioanna Papatsouma.pdf.

- Please note your coursework will be checked for plagiarism.

- Each piece of work must be accompanied by a signed coversheet.

- If possible, avoid last minute uploads as the system can crash if it simultaneously receives too many requests.

1. A car manufacturer, run by Mr. Ferrary, wishes to minimise its costs. Ferrary’s production process involves the use of raw materials (in quantity xR), labour (xL) and machinery (xM) to produce automobiles annually, according to the production function:

with a > 0 a constant.

(a) (4 marks) Compute the conditional factor demand functions ,  and , where wR, wL, wM > 0 are the input prices.

(b) (2 marks) Compute Ferrary’s cost function .

(c) (4 marks) Establish conditions on (α, β, γ) that will guarantee a profit-maximising position for Ferrary’s business, conditional on minimised costs.

(d) (2 marks) Using your answer from part (c), briefly describe any possible implications.

(e) (4 marks) Ferrary wishes to analyse the actions of another car manufac-turer, run by Ms. McLaurin. In the first week of the financial year, McLaurin produces and sells 400 cars at ↔40,000 each; in the following week, she produces and sells 500 cars at ↔35,000 each. Ferrary has heard that McLaurin increased her purchase of raw materials from 5,000 units in the first week to 7,000 units in the second; what can Ferrary deduce about the unit costs at which McLaurin was able to obtain raw materials in each of these weeks?

2. (4 marks) Consider a firm that produces a single output using two input factors. If the production function of the firm exhibits increasing returns to scale, show that a profit maximising bundle will typically not exist as long as there is some point that yields positive profit.