Quantitative Economics - ECON 500
Fall 2024
MIDTERM EXAM I
Instructions: The exam has two parts. Part A is in the form. of a multiple choice test, there is no partial credit in this section. Part B has questions that ask you to show your work, you can get partial credit in this section. You are allowed to pick the part that you like more. Your score will be equal to the highest of the two scores. You can try to do both parts, but there is no need to.
Part A
Problem #1
Let A denote the set of the algebraic numbers and T denote the set of transcendental numbers. Which of the following sets is finite?
a) (A ∪ T) ∩ Q
b) (R\T) ∩ T
c) Q∩A
d) T ∪ Q
Problem #2
Consider the following sets: A = {z ∈ C|z3 + 1 = 0} and B = 2A . How many elements does B have?
a) 8
b) 4
c) 2
d) ∞
Problem #3
Let f(x) = 1 − x. What is f
(2n+1)(x) = f°f°...°f(x)
a) x
b) 1 − x
c) (1 − x)
n
d) None of the above
Problem #4
Let f(x) = 1 − x for x ∈ R. What is the inverse of function of f, i.e., f-1 ?
a) It does not exist
b) 1 − x
c) 1-x/1
d) None of the above
Problem #5
Which of the following sequences converges?
a) an = e-sin(n2 π)
b) an = 1 + + + ... +
c) an = (−1)ne-n
d) an = (−1)nnsin(n/1)
Problem #6
Consider the following recursive structure Pt+1 = 20 − P. Imagine that P = λPt+1+(1−λ)Pt , where λ ∈ [0, 1]. For which values of λ convergence occurs regardless of the starting point?
a) Never
b) Always
c) for λ > 4/1
d) only for λ = 1
Problem #7
Consider the following sequences an = nsin( ), and bn = ), where b satisfies 2 − b = eb. What is the limit of cn = + bn?
a) It does not exist
b) 1
c) 2
d) ln(2)
Problem #8
Imagine that we have two vectors v = [cos2 (α), a, b,1] and w = [1, −b, a, sin2 (α)]. What is the angle between the two vectors?
a) 0
b) α
c) 2/π
d) None of the above
Problem #9
Consider the following sets B1 = {x|p ◦ x ≤ 48} and B2 = {x|q ◦ x ≤ 48}, where p = [1, 2] and q = [2, 1]. Which of the following sets is convex?
a) B1 ∪ B2
b) B1 ∩ B2
c) The boundary of B1 ∩ B2
d) None of the above
Problem #10
Consider the following preferences
Assume that xi ∈ {0, 1, 2, ..., 9} and yi ∈ {0, 1, 2, ..., 9} . Which of the following utility functions represents the above preferences?
a) U(x, y) = x + y + z
b) U(x, y) = 100x + 10y + z
c) Such a representation does not exist
d) None of the above
Problem #11
Let Abe an n × n symmetric matrix and x be an n × 1 vector. What are the dimensions of xTAx − (xTAx)T?
a) Such an object does not exist
b) n × n
c) 1 × 1
d) 0
Problem #12
Consider the following matrix
where d is the determinant of C = (cij) where cij = ei-j for i,j ∈ {1, 2} . Let bT = [1, 0, −1]. System Ax = b
a) has a unique solution
b) has no solutions
c) has infinitely many solutions
d) has only a trivial solution, i.e., only x = 0 solves the system
Problem #13
Let A be a matrix whose determinant is equal to a 0. Let C = AAT − (AAT)T . What is the determinant of C-1 ?
a) a2 − a-2
b) 1
c) 0
d) None of the above
Problem #14
Consider the following matrix
Its determinant of A2 is equal to
a) 0
b) 8
c) 64
d) None of the above
Problem #15
The inverse of C = AA-1 − (AA-1 )-1
a) does not exist
b) is the same as C
c) is equal to I
d) None of the above
Problem #16
Consider the following matrix
Its eigenvalues are given with
a) λ1 = 2 and λ2 = 4
b) λ1 = −2 and λ2 = 4
c) λ1 = 2 and λ2 = −4
d) None of the above
Problem #17
Imagine that λi ’s are non zero distinct eigenvalues of a given matrix, A, and vi’s are the corre- sponding eigenvectors. Let P be the matrix of eigenvectors and D be the diagonal matrix with eigenvalues on the main diagonal. What is the determinant of Ck , where C = AP − PD?
a) 0
b) λ1λ2...λk
c) λ1(k)λ2(k) ...λk(n)
d) None of the above
Problem #18
Consider the following system
Do you expect the system to converge to the steady state?
a) Yes
b) No
c) Only when |a| < 1
d) None of the above
Problem #19
Consider the following matrix
Let B = (PT P)N . We can be sure that the eigenvalues of B are
a) less the 1 in absolute value
b) both equal to 1
c) approach ∞ and -∞ as N becomes large
d) None of the above
Problem #20
Imagine that a given symmetric matrix A has distinct eigenvalues and has an eigenvector given with v1(T) = [1, 2]. Which of the following could be the other eigenvector of A?
a) v2(T) = [2, 2]
b) v2(T) = [-2, 1]
c) v2(T) = [-2, 2]
d) v2(T) = [2, 1]
Part B
1. The calibration of a macroeconomic model suggest the following recursive rule of motion for expected inflation:
πt(e) = 0.1 + 0.8πt-1 + ϵt
Further, assume that πt = πt(e), π0 is given, and ϵt = (-1)t × 0.01
(a) Find an expression for πt as a function of π0 and t.
(b) Verify your formula is correct using a proof by induction.
(c) Do you expect a sequence as defined in πt(e) – or the one you wrote for πt, if it is easier to analyze for you – would converge to a unique limit? Briefly explain why.
2. Prove the following:
(a) The set of complex numbers: C = {x = a + bi : i =
√−1, and a, b ∈ R} is convex.
(b) The set L = {(x, y) : x ∈ R ∧ y ∈ Z}, where Z are the integer numbers, is not convex.
3. Consider a n × n matrix A. The elements of this matrix have the following property: a2j = αa1j , ∀j Λ α ∈ R. Show that det(A) = 0.
4. Consider the following system of equations that describes a simple national-income model:
(a) Express the system in matrix notation: AX = B. Where
(b) Find the inverse of matrix A.
(c) What is the solution of the system if C0 = I0 = G0 = 1 and c = 0.5?
5. Consider the following matrix:
(a) Compute the eigenvalues of D.
(b) State the associated eigenvectors.