PART BPlease complete in BLOCK CAPITALS
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PART APlease complete clearly
Exam Number
as shown on your university card
Advanced Mathematical Economics
ECNM10085
Wednesday 20 December 2023
13:00:00–16:00:00
Number of questions: 9
Total number of marks: 100
IMPORTANT PLEASE READ CAREFULLY
Before the examination
1. Put your university card face up on the desk.
2.Complete PART A and PART B above. By completing PART B you are accepting the
University Regulations on student conduct in an examination (see back cover).
3. Complete the attendance slip and leave it on the desk.
4. This is a closed-book examination. No notes, printed matter or books are allowed.
5. A calculator is permitted in this examination. It must not be a programmable or graphic
calculator. It must not be able to communicate with any other device.
During the examination
1. Write clearly, in ink, in the space provided after each question. If you need more space then
please use the extra pages at the end of the examination script or ask an invigilator for
additional paper.
2. Your exam will be marked on fundamentals (45 points), model formulation (10 points), tools
(10 points), and logic (35 points).
3.You should answer all of Section A and only some of Section B..
4. If you have rough work to do, simply include it within your overall answer – put brackets at the
start and end of it if you want to highlight that it is rough work.
At the end of the examination
1. This examination script must not be removed from the examination venue.
2. There are extra pages for working at the end of this examination script. If used, you should
clearly label your working with the question to which it relates.
3. Additional paper and graph paper, if used, should be attached to the back of this examination
script. Write your examination number on the top of each additional sheet.
[Do not write on this page]
ECNM10085Do not write above this lineDecember 2023
Section A
1. Suppose an engineering firm designs and builds apartment buildings. It hires both full-time and
part-time workers. Assume that full-time workers are more productive, because they complete
urgent tasks more quickly, and are easier to reach to resolve problems. Assume that all households
have two workers, and some households have children. Households with children have a stronger
preference for part-time work. Households own the engineering firm, supply labour, and buy
homes.
(a) Formulate a competitive model of the three markets (the market for apartments, and full-time
and part-time labour markets).
(b) Prove that if the wages of part-time workers increases, then firm demands fewer part-time
workers.
(c) Reformulate the firm’s problem with a Bellman equation in which the only choice is the
amount of apartment construction.
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Section B
2. (easy) Provide a counter-example to the following false claim: If (X,d) is a metric space, and the
interior ofA?Xis connected, thenAis connected.
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4. (easy) Pick any setAinside a metric space (X,d). Pick any radiusr >0 and letU={x: (x,a)∈
X×A,d(x,a)< r}be the set of all points inXthat have a distance of less thanrto some point
insideA. Prove thatUis an open set.
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5. (easy) Suppose there are two bidders in an auction for the remnants of a bankrupt car factory.
The first bidder values the factory at￡20m. The first bidder spied on the second bidder, and
knows he will bid￡10m. Thus, his (expected) profit when biddingbmillion is
Calculate the rangeπ(R), the maximum maxπ(R) and the supremum supπ(R), or prove that
they do not exist.
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6. (medium) Suppose that (X,d) is unbounded. Prove that there is a continuous functionf:X→R
that does not have a maximum.
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7. (medium) Suppose (X,d) is a disconnected metric space. Prove that there is a continuous function
f:X→Xthat does not have any fixed point, i.e. there is nox
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8. (medium) Consider a contractionf:X→Xof degreekon the metric space (X,d). Let.
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9. (hard) Suppose thatA?UandB?Vare non-empty sets, andUandVare disjoint open sets,
and all four sets lie inside the metric space (X,d). Prove thatA∪Bis disconnected.
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relates, otherwise your working may not be marked.
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relates, otherwise your working may not be marked.
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relates, otherwise your working may not be marked.
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relates, otherwise your working may not be marked.
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Exam Hall Regulations
The following is a copy of a Notice which is displayed in Edinburgh University Examination Halls for the
information of students and staff.
The University of Edinburgh Exam Hall Regulations
1. An examination attendance sheet is laid on the desk for each student to complete upon arrival. These
are collected by an invigilator after thirty minutes have elapsed from the start of the examination.
Students are not normally allowed to enter the examination hall more than thirty minutes after the start
of the examination.
2. Students arriving after the start of the examination are required to complete a “Late arrival form” which
requires them to sign a statement that they understand that they are not entitled to any additional time.
Students are not allowed to leave the examination hall less than thirty minutes after the
commencement of the examination or within the last fifteen minutes of the examination.
3. Books, papers, briefcases and cases must be left at the back or sides of the examination room. It is an
offence against University discipline for a student to have in their possession in the examination any
material relevant to the work being examined unless this has been authorised by the examiners.
4. Students must take their seats within the block of desks allocated to them and must not communicate
with other students either by word or sign, nor let their papers be seen by any other student.
5. Students are prohibited from deliberately doing anything that might distract other students. Students
wishing to attract the attention of an invigilator shall do so without causing a disturbance. Any student
who causes a disturbance in an examination room may be required to leave the room, and shall be
reported to the University Secretary.
6. Personal handbags must be placed on the floor at the student’s feet; they should be opened only in full
view of an invigilator.
7. An announcement will be made to students that they may start the examination. Students must stop
writing immediately when the end of the examination is announced.
8. Answers should be written in the script book provided. Rough work, if any, should be completed within
the script book and subsequently crossed out. Script books must be left in the examination hall.
9. During an examination, students will be permitted to use only such dictionaries, other reference books,
computers, calculators and other electronic technology as have been issued or specifically authorised
by the examiners. Such authorisation must be confirmed by the Registry.
10. The use of mobile telephones is not permitted and mobile telephones must be switched off during an
examination.
11. It is an offence against University discipline for any student knowingly
?to make use of unfair means in any University examination
?to assist a student to make use of such unfair means
?to do anything prejudicial to the good conduct of the examination, or
?to impersonate another student or allow another student to impersonate them
12. Students will be required to display their University Card on the desk throughout all written degree
examinations and certain other examinations. If a card is not produced, the student will be required to
make alternative arrangements to allow their identity to be verified before the examination is marked.
13. Smoking and eating are not allowed inside the examination hall.
14. If an invigilator suspects a student of cheating, they shall impound any prohibited material and shall
inform the Examinations Office as soon as possible.
15. Cheating is an extremely serious offence, and any student found by the Discipline Committee to have
cheated or attempted to cheat in an examination may be deemed to have failed that examination or
the entire diet of examinations, or be subject to such penalty as the Discipline Committee considers
appropriate.
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