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Homework 1

ECON6003/6703 Mathematical Methods for Economics

Note. This ﬁle contains the questions for Quiz 1. Work on your answers. Then submit your answers to these questions through the Canvas Quiz portal for Q1 - Q9. The last question, Q10, requires a PDF upload,via a diﬀerent portal. Everything is accessible from the Assignments tab on Canvas.

Submission Deadline: Midnight, Monday, 27 March

Quiz weight. 10% of the Final Grade.

If you do not submit on time. Your In-semester Exam will count for 45% of Final Grade.

Important. Once you submit your answers, your score for stands. You cannot then opt to transfer the score to the In-Class assignment (if you are unhappy with your performance here).

INSTRUCTIONS

1. Q1-Q9 Unlike the usual MCQ you may be accustomed to, in the following questions, there is not necessarily a unique correct choice — multiple options may be correct. For questions with multiple correct answers, total points for a question are divided equally between each correct selection. I deduct points if you are over-selecting answers. For example: To illustrate, suppose (a) and (b) are the only two correct answers to a 10 point question.

your choice mark Explanation
a,b	10	Full mark
a,b,c	5	(5 deducted for choosing (c), since )
		you choose three options when only two are correct
a,d	5	no deduction for choosing the incorrect (d),
		since only two options are chosen

2. In many cases, you should be able to get at the answer by considering simple examples / counterexamples to the given statements.

3. Each of the TEN questions is worth 10 points.

QUESTIONS

Question 1. Three friends share an house. They are looking to buy furniture for the common area. A, B and C are the sets of furniture bundles that the three friends respectively (weakly) prefer to the currently in their house. The new furniture must be weakly preferred by at least two of the friends. The furniture bundles feasible for the three friends is

a) (A\B)[(B \C)\(A\C)

b) (A\B)[(A\C)[(B \C \Ac)

c) (A\B \Cc)\(A\Bc \C)[(Ac \B \C)

d) (A\B \Cc)[(A\Bc [C)[(Ac \B \C)

Question 2. Consider the following statement:

Suppose A1 ;:::; An ;:::; is an infinite sequence of sets. To ensure that the ........B..1......

of this inhnite sequence is .....B..2. we need to assume that each An is ....B..3.. for every n.

Use the words “intersection, union, closed, open” and ﬁll in the blanks B1, B2, B3 to make the above a true statement. (Those words may be used multiple times.)

Question 3. Let S be a subset of real numbers. Which of the following statements are true?

a) If S 仁 [0; 100] then S is compact.

b) If S is closed, then S \[0; 100] is closed

c) If S contains an open subset, then S is open.

d) None of the above.

Question 4. Which of the following are true statements:

a) For an inﬁnite sequence an , if limn!1 an = a, where a is a ﬁnite number, then the set S ={a ; a1 ; a2 ; ::: ; g is compact.

b) A sequence an may have sub-sequences that converge, even though an itself does not converge.

c) A sequence an may have no convergent sub-sequence.

d) If an +bn converges, then each of the sequences an and bn must also converge

Question 5. The following sequence

0; 1; 0; ; 0; ; 0 : : : ; 0; ; 0; : : : is

a) bounded

c) converges to 0.

d) None of the above.

Question 6. Which of the following is the only true statement?

a) If f + g is continuous, then f or g must be continuous.

b) If an !A and an +bn ! L, then bn must converge.

c) If S1 and S2 are open sets, then S1 \S2 is neither open nor closed.

d) None of the above.

Question 7. Let f:S ¡!R be such that there exist a ; b 2S such that f(a) <0 and f(b) > 0. Claim: There exists c 2 S such that f(c) =0. The Claim is true if

a) If f is continuous.

b) If S is compact.

c) If S contains all its interior points.

d) None of the above.

Question 8. There are two markets. In market A the proﬁt of a certain ﬁrm depends on its price and that of its competitor in market A. If the competitor's price is ﬁxed at q , proﬁt of this ﬁrm is

8 (10¡ p) p if p < q

f(p) = K if p = q

( 0 otherwise:

where K is some ﬁxed constant. Pick p0 such that 0 < p0 < 10.

a) f must be continuous at p0 .

b) f is continuous at p0 if p0 = q .

c) f is continuous at p0 if p0 q .

d) f is necessarily continuous if we set K = f(p0)

e) f may not be continuous but its limit is well-deﬁned for any p0 2 (0; 10)

Question 9. A standard utility maximizing consumer with preferences over two goods is known to receive quantity discounts on the ﬁrst good leading to a budget set is as shown in the diagram below the yellow shaded region including the edges.

Figure 1.

Moreover, his utility for any consumption bundle x = (x1 ; x2) is given by u(x) = min { ; }. In this setup we cannot use the MRS=price ratio formula to ﬁnd the utility maximizing bundle (why?). Fill in the following blank labelled B to make the statement below a true statement.

A utility maximizing consumption bundle exists by appealing to B Theorem.

Note: Simply ﬁll that blank with “NO” if you think that it is possible that a utility maximizing choice may not exist in the above environment.

Question 10. This question requires you to write down a formal proof. Mimic the way proofs are written in text books/problem set solutions etc. and upload a handwritten answer as a single PDF ﬁle on Canvas Quiz Portal.