SUMMER TERM 2024
DEPARTMENTALLY MANAGED REMOTE ONLINE EXAMINATION
ECON0006: INTRODUCTION TO MATHEMATICS FOR ECONOMICS
Assessment Component: 80% Remote Online Controlled Condition Examination
Time Allowance: You have 2 hours to complete this examination, plus an additional collation time of 20 minutes and an Upload Window of 20 minutes. The additional collation time has been provided to cover any additional tasks that may be required when collating documents for upload, and the Upload Window is for uploading and correcting any minor mistakes. The additional collation time and Upload Window time allowance should not be used for additional writing time.
If you have been granted SoRA extra time and/or rest breaks, your individual examination duration and additional collation time will be extended pro-rata and you will also have the 20- minute Upload Window added to your individual duration.
If you miss the submission deadline during the 40-minute Late Submission Period and do not receive approved mitigation for the circumstances relating to your late submission, a Late Submission Penalty will be applied by the Module Administrator. At the end of the Late Submission Period, you will not be able to submit your work via Moodle and you will not be permitted to submit work via email or any other channel.
All work must be submitted anonymously in a single PDF file. Do not write your name and student number in either the file or the file name. The file name must be in the following format: Module Code-%Exam i.e. ECON0006-80%Exam.
Page Limit: 24 pages.
Academic Misconduct: By submitting this assessment, you are confirming that you have not violated UCL’s Assessment Regulations relating to Academic Misconduct contained in Section 9 of Chapter 6 of the Academic Manual.
Number of Questions Answered Policy: In cases where a student answers more questions than requested by the examination rubric, the policy of the Economics Department is that the student’s first set of answers up to the required number will be the ones that count (not the best answers). All remaining answers will be ignored.
QUESTIONS
Answer ALL FIVE questions from Part A and TWO questions from Part B.
Questions in Part A carry 10 per cent of the total mark each and questions in Part B carry 25 per cent of the total mark each.
PART A
1. Find the values ofa for which the matrix
is not invertible.
For all other values ofa find an expression for the inverse.
2. Find the range of values ofk for which the quadratic forms
are positive definite.
3. Suppose
Z = y3 exy.
Use the small increments formula to find, in terms of e, an approximation to the change in z when x changes from 1 to 1 + Δx andy changes from 1 to 1 + Δy, where Δx and Δy are small.
Now suppose you are additionally given
x = 2t, y = t 2 + t.
Use the chain rule to find, in terms of e, dZ/dt when t = 1.
4. Find the x andy coordinates of the critical points of the function
f (x, y) = 3x2 − y3 +12xy − 36y
and classify each critical point as a local maximum, local minimum, saddle point or none of these.
5. Find the general solution of the difference equation
3y t+1 + 2y t = 3.
What happens to the general solution as t → ∞? Find also the solution which satisfies y = 1 when t = 0 and sketch its graph.
PART B
6. Find the determinant of the matrix
Solve the equation system
x + 2y + λz = 0
2x + 3y — 2z = λ
λx + y + z = 3
for all values of λ .
7. (a) Suppose the production function in an economy takes the form.
F(K, L) = KαLβ (α, β > 0)
where K andL denote capital and labour respectively.
Find a condition in terms of α and β for the production function to exhibit decreasing returns to scale. Show that if this production function exhibits decreasing returns to scale, then it also exhibits diminishing returns to each input. State, with reasons, whether the converse is true.
(b) Suppose F(K,L) is a general production function satisfying F(0,0) = 0 . Show that if F(K,L) is concave then it cannot display increasing returns to scale.
(c) Now suppose the production function in an economy takes the form. Q = H(K, L)ert
where Q, K, L and t denote aggregate output, capital, labour and time respectively. Suppose further that r is a positive constant, H(K, L) is homogeneous of degree s where s is a positive constant and K and L have the same constant proportionate rate of growth m. Find the rate of growth of output.
8. Consider the production function
where Q, K and L denote output, capital and labour respectively and where a and b are positive constants.
Sketch the isoquant diagram.
Suppose that the prices of capital and labour are r and w respectively. Find the cost function.
Find also the elasticity of substitution.