.    Consult MFE Assessment Criteria nearby, for a summary of what I and (probably) other markers of mathematical and quantitative methods in all years are looking for.

.    Consult MFE Mathematical  Writing And Explanation nearby, for how to get  most of the 50  marks available for good  explanation.  Pay attention to the advice to  use a ‘running commentary’ style of explanation.

.    The answers in Problems and Answers and Seminar Answers provide a useful guide.

.    The  reason for these requests is that the module stresses communication as well as mathematical skill. It is of little use knowing the maths, if you can’t communicate it to someone else.

.    About some of these things, you should make your own judgement, rather than asking us what is the ‘right’ thing to do. Making such judgements is part of what you are marked on. Credit will be given for a reasonable amount of self-reliance, as it will in your future career.

.    Use the Hints.

Feedback to Students:

Both Summative and Formative feedback is given to encourage students to reflect on their learning that feed forward into following assessment tasks. The preparation for all

assessment tasks will be supported by formative feedback within the tutorials/seminars.   Written feedback is provided as appropriate.  Please be aware to use the browser and not the Canvas App as you may not be able to view all comments.

Because we must mark and provide feedback for hundreds of students within 15 working days, and treat students consistently, we don’t provide in our feedback verbal comments and corrections for each individual answer.

Instead, we provide:

.    a mark for each question.

.    specific feedback comments to you.

.    general  feedback comments to the entire group,  including an account of common errors.

.    Contact details in case there’s anything you’d like to discuss.

Plagiarism:

It is your responsibility to ensure that you understand correct referencing practices. You are

expected to use appropriate references and keep carefully detailed notes of all your  information sources, including any material downloaded from the Internet. It is your responsibility to ensure that you are not vulnerable to any alleged breaches of the     assessment regulations. More information is available at https://intranet.birmingh am.ac.uk/as/studentservices/conduct/misconduct/plagiarism/index.aspx.

Physical Aspects

.    Please submit all answers in a single pdf file. Do not use other formats, such as docx or png. An Assignment Box will be made available for this purpose.

.    Scanned, hand-written answers are permissible. You may mix in the same file pictures taken by phone or scanner, and typed text.

.    In the interests of legibility, any photographs of text should be full-on, not from an angle. Please check for legibility before submitting.

.    For both our sakes, please write your answers clearly and make them easy for us to read and mark. Ideally, prepare a rough version first, but submit a fair copy. If your   answer occupies several pages, look for a more concise version, which will probably gain more marks. The assignment is supposed to test mathematical understanding

and the ability to explain mathematical ideas, not stamina.

.    Please include page numbers and question numbers.

.    Sets of answers that do not conform to these requests will be penalized according to how much difficulty they create for the markers.

.    Answers received after the deadline, even if submission began before the deadline, count as late, and are subject to the usual 5-marks-per-day penalty. Why not submit a day or two before the deadline?

.    A submission counts as late if it is recorded as late by Canvas. What counts is when the submission process is complete, not when it starts. I advise you to submit your  answers a few hours ahead of the deadline, at the latest.

.    If the Canvas time is given as one minute after the deadline, the penalty still applies. If you miss a bus by one minute, you miss it.

.    If you decide to re-submit your answers (perhaps because you have noticed an error) it is the time of re-submission that counts, not the time of your first submission. So if you re-submit a corrected version after the deadline, your submission counts as late.

MFE 50% Assignment Semester 1, 2023-24 Session

Q1 and Q2 - Week 1. From Pure Maths To The Maths Of Economics And Finance

Q1        Simplify exp (2 lnA + 3 ln B − 4 ln C + 5 lnD ), where A, B, C, D are all positive quantities.

Q2        Revenue R(P, Q) = PQ is a function of two variables: price P and quantity Q. Sketch a contour map of R in the region 0 ≤ P ≤ 10, 0 ≤ Q ≤ 20.

HINTS: (i) What mathematical shape is exemplified by the equation xy = 12? What would a contour map of the function f(x, y) ≡ xy look like?

(ii)         One suitable contour height for R(P, Q) is ℎ = 50. Pick other contour heights using your own judgment.

(iii)        Note that the question asks for a sketch, not an exact plot.

Q3 and Q4 - Week 2: Things Economists Do With Differentiation

Q3        (a)         By gathering berries for t hours, Robinson Crusoe can obtain X(t) kg of

berries, where X(t) = 5√t. How many kg of berries will he obtain in his fifth hour of labour?

(b)         Robinson works continuously, starting at 8:00 am. At noon, what is his marginal product of labour-time?

Q4        (a)          National income Y currently stands at $20 tn/yr (trillion dollars per year) and

consumption C = C(Y) at $16 tn/yr. If marginal consumption C ᇱ (Y) = 0.7, use the Small Increment Formula to obtain an approximation to the level of consumption if Y rises by $0.2 tn/yr.

(b)        The general price level P is rising over time according to the formula P(t) = P (exp(at) + bt), where P, a, and b are positive parameters. Calculate:

(i)         The rate of increase of prices, P(̇) at t;

(ii)        The rate of growth of prices at t. Then

(iii)        Evaluate both P(̇) and when t = 0.

Q5 - Week 3: Unconstrained Optimization With A Single Choice Variable

Q5        Consider the optimization problem whose formal statement is

min f(x), where f(x) ≡ 3xସ − 4xଷ − 12xଶ + 10  [P]

௫

(a)        Write down the first-order condition for this problem.

(b)         Find any critical points of the problem.

(c)         By using the second derivative function f ᇱᇱ (x), decide which critical points, if any, are strict local minima off(x).

(d)         Using the information obtained so far, solve the problem P.

Q6 and Q7 - Week 4: Many Variables. Constrained Optimization.

Q6        (a)          Mr Jones has utility function U(A, B) = AB. He loses b bananas, reducing his

banana holding to B − b. How many extra apples a will he need as compensation, to restore him to his former level of utility?

(b)         From consuming A apples and B bananas, Ms Jones gets utility U(A, B) = √AB + A + 2B. What is Ms Jones’ marginal utility with respect to apples, in terms of A and B ?

Q7        My utility function over apples (A) and bananas (B) is U(A, B) = Aଶ B. If apples cost $6 per kg, and bananas cost $4 per kg, and I have $50 to spend. I wish to maximize  utility, subject to the constraint that the value of my purchases equals by budget      limit $50.

(a)         State my problem formally, as a problem of constrained optimization, and say

what the objective function and choice variables are.

(b)        Write down the corresponding Lagrangean function.

(c)         Obtain first-order conditions, and state what other condition is needed to obtain a solution.

(d)        Obtain optimal values A∗, B ∗, explaining your working clearly. You may

assume that A∗ ≠ 0 at the solution. You may also assume that the second- order conditions for this constrained problem are satisfied.

Q8 and Q9 - Week 5: Logic, Sets, and Functions.

Q8        (a)         Express the set x: x ∈ ℝ(ା)  ∧ (xଶ  < 16) using interval notation.

(b)        Are the following three statements logically equivalent? Use the symbols and methods of propositional logic to find out, and explain.

i.       Without a well-educated labour force, the economy cannot thrive. ii.       If there is a well-educated labour force, the economy can thrive.

iii.       If the economy can thrive, the labour force must be well-educated.

HINT. Let e mean ‘The workforce is well-educated’.  Let t mean ‘The economy can thrive’ .