Controlled Condition Exam: 1 Hour exam
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module in the following academic year(s):
Year
2020/21 & 2021/22
Additional material
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Exam paper page
limit
12 pages
TURN OVERECON0010
1
TURN OVER
SUMMER TERM 2022
CENTRALLY MANAGED ONLINE EXAMINATION
ECON0010: MATHEMATICS FOR ECONOMICS
Time Allowance: You have 1 hour to complete this examination, plus an Upload Window of 20 minutes. The
Upload Window is for uploading, completing the Cover Sheet and correcting any minor mistakes and should
not be used for additional writing time.
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extended pro-rata and you will also have the 20-minute Upload Window added to your individual duration.
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an online examination in the AssessmentUCL Guidance for Students.
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will not be permitted to submit the work via email or any other channel. If you are unable to submit your
work due to technical difficulties which are substantial and beyond your control, you should apply for a
Deferral via the AssessmentUCL Query Form.
Page Limit: 12 pages. Your answers, excluding the Cover Sheet, should not exceed this page limit. Please
note that a page is one side of an A4 sheet with a minimum margin of 2 cm from the top, bottom, left and right
borders of the page. The submission can be handwritten or typed, but the font size should be no smaller than
the equivalent to an 11pt font size. This page limit is generous to accommodate students with large
handwriting. We expect most of the submissions to be significantly shorter than the set page limit. If you
exceed the maximum number of pages, the mark will be reduced by 10 percentage points, but the penalized
mark will not be reduced below the pass mark and marks already at or below the pass mark will not be reduced.
Answer 1 question from Part A and 1 question from Part B.
Questions in Part A carry 50 per cent of the total mark each and questions in Part B carry 50 per cent of
the total mark each.
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.
If you have a query about the examination paper, instructions or rubric, you should complete an
AssessmentUCL Query Form. Please note that you will not receive a response during your examination.
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.ECON0010
2
CONTINUED
PART A
Answer 1 question from this section.
A1.
(a) Find the critical points of the function
.
Determine the nature of each critical point.
(b) Show that the function
𝑥 4 + 𝑦 4 − 𝑥 2 − 2𝑥𝑦 − 𝑦 2
has a critical point at (0,0,0) and determine whether this point is a local maximum, a
local minimum, a saddle point or none of these.
A2.
In the following market model, is sales, is output and
is price, each in period t.
𝑠𝑡 = 5 − 𝑝𝑡
(1)
𝑞𝑡 = −3 + 11𝑝𝑡−1 (2)
𝑝𝑡 − 𝑝𝑡−1 =
1
4
(𝑠𝑡 − 𝑞𝑡) (3)
Obtain a first-order difference equation for p. Find the stationary solution and show that it
is unstable.
(a) Suppose (2) is replaced by
𝑞𝑡 = −3 + 𝛼𝑝𝑡−1
where 𝛼 is a positive constant. Find the stationary solution and the range of values of 𝛼
for which the stationary solution is stable.
(b) Suppose instead (3) is replaced by
𝑝𝑡 − 𝑝𝑡−1 = 𝛽(𝑠𝑡 − 𝑞𝑡)
where 𝛽 is a positive constant. Find the stationary solution and the range of values of 𝛽
for which the stationary solution is stable.
(c) What can you say about the situation where both of the replacements specified in (a)
and (b) are made at the same time?
y 2 (1− x) − x 2 (1+ x)
st
qt
ptECON0010
3
END OF PAPER
PART B
Answer 1 question from this section.
B1.
A consumer has a Stone-Geary utility function
where xi denotes the consumption of the i-th commodity and b1,b2,c1 and c2 are positive
constants. The price of the i-th commodity is pi and the consumer’s income, m, is such
that
(*) .
Show that each indifference curve is negatively sloped, convex and has the lines x1=c1 and
x2=c2 as asymptotes. Sketch the indifference curve pattern.
Express the consumer’s problem as a constrained maximisation problem. Explain the
significance of the condition (*).
Explain with the aid of a diagram how the indifference curve pattern is modified when
b1,b2 and c2 are positive but c1 is negative.
In the case 𝑐1 = −2, 𝑐2 = 3, find the demand functions which are applicable whenever
𝑚 > 3𝑝2.
B2.
(a) Suppose that x and y satisfy the following system of differential equations
2y x cos t,
x
+ − =
y −2x − y = sint .
Find the second-order differential equation satisfied by the complex variable
.
Find the general solution of this differential equation for all real values of
. Hence
find, for all real values of
, the solutions satisfying the conditions x = x = y = y = 0
at
.
(b) For the system of differential equations
find the stationary solution and sketch the phase diagram.
Verify algebraically that the stationary point is a saddle point and find the equation of the
stable branch
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