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ECON2125/6012, Semester-1 2023 Mid-Term Exam

发布时间：2023-06-01

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Mid-Term Exam

ECON2125/6012, Semester-1 2023

1. Duration: 10 min Reading + 120 min Writing + 15 min Submission: from 6:40PM to 9:05PM.

2. Exam is open book. You can use your course materials but you are not allowed to ask for hints or answers from any website or any person.

3. You are required to keep your camera on during the entire exam duration (6:40PM-9:05PM). Your camera must face you.

4. Submit your exam answers via the submission link named “Mid-Semester Exam Submission Link” in the ‘Mid-Semester Exam’ block. Once you have submitted your work,

a. Check Turnitin to make sure that you can see your submission there.

b. Check your email and make sure you have received a successful submission confirmation email from Turnitin. It is your responsibility to ensure that your submission is

successful.

5. Your answers must be hand-written (either on papers or tablet such as iPad).

6. You must submit your exam answers in ONE file ONLY and in PDF format ONLY.

7. You can use Microsoft Lens app or you can take photos of your exam answers, copy the image files into a word document, and use the ‘save as…’ function to save the word document as a PDF file.

8. Make sure your PDF file is readable, and that your scanned answers are legible.

9. Make sure you upload the correct exam file.

10. The file name of your exam should follow the format: Surname_UniID (e.g., Keynes_u1234567; Hajargasht_u9876543).

The Exam

• Answer all questions.

• Write down all of your working.

Question- 1:

Part-A: Consider S =〈 1+ | n = 卜恳1} 仁 . Determine if

(i) S is open 12 marks

(ii) S is closed

(iii) S is compact

(iv) S is connected

(v) S is a complete subspace

(vi) S is convex

Justify your answer with a short verbal or mathematical argument.

Part-B: Given any two points x = (x1 , x2 ) and y = (y1 , y2 ) in R2, define the function

f (x, y) = x1 − y1 + | y2 − x2 | . Is this a metric in R2 ? Prove your answer. 6 marks

Part-C: Let S = {x | x is irrational and 0 共 x 共 5} {x | x is irrational and 6 共 x 共 10} . Is this

an open or closed set in R? Provide an argument for your answer. 7 marks

Question-2:

Part-A: Consider an economy which can be in one of the two states of being in recession (R) or in expansion (E). Assume that the state of the economy in each period depends on its

previous period with probabilities P(Et+1 | Et ) = 0.9 and P(Rt+1 | Rt ) = 0.4 . 12 marks

(i) Write down the equations describing the dynamics of the model.

(ii) Solve the resulting difference equation.

(iii) What is the chance of a recession in the long-run.

Part-B: Let Q = 3x1(2) + 4x2(2) + 6x3(2) + 2 x2x3 . Determine the positive or negative definiteness

of this quadratic form. Write down all of your working. 6 marks

Part-C: Prove that (Xk，根n Λ n(−)根(1)n Xn根k )−1 is positive definite if Λ is an invertible symmetric

positive semidefinite matrix. Assume all inverses exist. 7 marks

hint: inverse of a positive definite matrix is positive definite.

Question-3:

( N N )

Part A: Consider the cost function C(w, y) = |xxaij |y with aij = aji and where w

is the vector of input prices and yis the output. 18 marks

(i) Under what conditions on the coefficients aij s, C(w, y) is homogenous of degree one with respect to w?

N a ln C

i=1 a ln wi

(iii) Find the Hessian of C at point w = (1, ...,1) and y = 1 .

Part B: Consider the macroeconomic model defined by system 7 marks

Y = C + G + I C = f (Y −T) I = h(r) r = m(M )

where f , g , h are C1 functions. Give sufficient conditions for the system to determine Y,C, I andras a function of G,Tand Min the neighbourhood of an equilibrium point.

Question-4:

Part A: 10 marks

(i) Find stationary points of f (x, y, z) = (x2 + 2y2 + 3z2 + 2xy + 2xz)3

(ii) Determine the type of stationary points.

Part B: Consider matrices yN根1 , XN根K , βK根1 , N = N1 + N2 and

15 marks

f (β,o1(2) ,o2(2)) = exp〈− [y − Xβ]T Σ −1 [y − Xβ ]卜 where Σ = 2o10N2I根N1(N1) 0N12o2根IN(N)2(2)

(i) Find stationary point of this function with respect to (β,o1(2) ,o2(2)) . (ii) Is this point a maximum or a minimum? Prove your answer.

I is the identity matrix. Assume all relevant inverses exist.