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]

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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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