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ECON6003/6703 Mathematical Methods for Economics Homework 2

发布时间：2023-05-13

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Homework 2

ECON6003/6703 Mathematical Methods for Economics

Note. This ﬁle contains the questions for Quiz 2. Work on your answers. Then submit your answers to these questions through the Canvas portal. The speciﬁc link will be sent as a Canvas announcement later during this week.

Submission Deadline: Midnight, Monday, 15 May, 2022.

Quiz weight. 10% of the Final Grade.

If you do not submit on time. Your Final Exam will count for 55% of Final Grade.

Important. Once you submit your answers, your score for the Quiz stands. The Quiz weight will then not be transferred to the Final Exam.

INSTRUCTIONS

1. 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 cor- rect. 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. Not every question has multiple correct choices – if you believe there is a unique correct choice, then go ahead and pick just that one.

3. Work through Problem Sets 4, 5 6 carefully – many (not all) questions here are related to those.

4. Each of the questions is worth 10 points.

QUESTIONS

Question 1. Consider the hyperplane H(p, 0) = {x ∈ R | p3 · x = 0}. The angle between any x∈ H(p, 0) and p equals A .

Enter the value of A in degrees on Canvas.

Question 2. Consider the set V ={v1 , v2 , v3} where

v1 = v2 = v3 = .

The linear span, L(V), of the above set is A R3 since the vectors in V are linearly B .

Choose the correct values for A and B from the choices (a)-(d) below that would make the above a “true” statement; and if none of them are work, choose (e).

a) A = “not equal to” , = “independent”

b) A = “not equal to” , B = “dependent”

c) A = “equal to” , B = “dependent”

d) A = “equal to” , B = “independent”

e) None of the above.

Question 3. Consider the Figure below.

Figure 1. Figure for MCQ (3).

a) The Separating Hyperplane Theorem does not apply for the set C .

b) There is a hyperplane that separates the point P from the set C .

c) Both (a) and (b).

Question 4 . Consider a two agent two good endowment economy in which the preferences of the two agents are described by the indifference curves depicted in the figure below .

Figure 2. Preferences of agent 1 and 2 on left and right panel resepectively.

The proof of the Second Welfare Theorem presented in class cannot be applied to the above economy because A Theorem cannot be applied.

Replace A with the name of the appropriate theorem.

Question 5. Consider the system of equations

''''「(l) lIIII(」) = .

This system has a non-negative solution in x. Choose “True” or “False” on Canvas. Hint: Use Farkas Lemma.

Question 6. For a certain function f : R → R that is diﬀerentiable everywhere, it is known that f(0) =3, f \ (0) = 1 and that the function is non-decreasing on [0, 10]. From this, we can conclude that for all x ∈ [0, 10],

a) f(x) ≥ x − 3.

b) f(x) ≥ 3 − x.

c) f(x) ≥ 3 + x.

d) None of the above.

Hint: Try applying the mean value theorems.

Question 7. Consider the following maximization problem with equality constraints

max F(x , y , z)

(x , y ,X ) ∈R3

s.t. h1 (x , y , z) = a

h2 (x , y , z) =b

where F , h1 and h2 are all diﬀerentiable. At a certain point (x* , y * , z *), the gradients of the three functions were computed to be:

∇F(x* , y * , z *) = ∇h1 (x* , y * , z *) = ∇h2 (x* , y * , z *) =

Pick the true statement(s).

a) (x* , y * , z *) is a solution to the maximization problem.

b) (x* , y * , z *) is a possible solution to the maximization problem.

c) (x* , y * , z *) cannot be a solution the maximization problem since the NDCQ is not met.

d) (x* , y * , z *) cannot be a solution the maximization problem ∇F(x* , y * , z *) does not lie in the linear span of ∇h1 (x* , y * , z *) and ∇h2 (x* , y * , z *).

e) None of the above.

Question 8. For a certain 2× 2 matrix A= (c(a) b(c) ), it is known that a<0 and b<0. Claim : Therefore, A is negative deﬁnite.

State whether the Claim is True or False.

Hint: You will have an immediate answer if you have worked through Prob Set 6.

Question 9. For a certain matrix, A = (c(a) b(c) ), The ﬁgure below shows a sample plot of the function Q(x) =x\Ax from two diﬀerent perspectives.

Figure 4.

This would be true if A satisﬁes the following conditions.

a) a < 0 and ac − b2 > 0.

b) a < 0 and ab − c2 < 0.

c) a > 0 and ab − c2 < 0.

d) None of the above.

Question 10. Let A be a n × n matrix with strictly positive entries and let ∆ = {x=(x1 , . . . , xn) ∈ R|对i xi =1}.

Now, consider the function f(x) = 之i(1)xiAx. By applying the A Theorem, we can conclude that the matrix A has an real eigenvalue λ* where λ* = B .

Fill in the blanks A and B.