Second Semester Examination– November 2022
OPTIMIZATION FOR ECONOMICS AND FINANCIAL
Copyright By PowCoder代写 加微信 powcoder
Reading Time: 0 Minutes
Writing Time: FOUR Hours
Permitted Materials: Any material
Page 1 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
Answer all questions in this section using the answer booklet(s) provided.
Answers are expected to be succinct but complete. Answers that are too long
and irrelevant will be penalized.
1. There are 55 marks of questions deemed straight forward, 25 marks deemed medium
and 20 marks deemed challenging.
2. Do not spend too much time on any single question. If you find helpful, you may
skip parts that look challenging to you first and work on them after you finish with
the easier parts.
3. Important Notice: Throughout the exam, the symbol “log” means natural loga-
Question 1 [15 marks] If there are two consistent estimators θ̂(1) and θ̂(2) of the same
n-dimensional parameter θ in an econometric model, with covariance V (1) and V (2), re-
spectively, then Estimator 1 is called weakly better than Estimator 2 if aTV (1)a ≤ aTV (2)a
for every a ∈ Rn. (Ignore the word “consistent” if you do not know what if means and
assume that all estimators in this question are consistent.)
1. [5 marks] (SF) Explain the significance of the expression aTV (1)a.
2. [10 marks] (SF) Now consider two consistent estimators θ̂(1) and θ̂(2) with
3 1 −11 5 2
, and V (2) =
2 0 −10 4 2
Determine whether Estimator 1 is weakly better than Estimator 2, the other way
around, or neither is weakly better.
Page 2 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
Question 2 [15 marks] Each part defines a subset A of R3 and a function f on A. First,
determine whether A is a convex set. If A is indeed a convex set, determine whether f is a
concave function, a convex function, or neither. (In case a function is concave or convex,
there is no need to determine whether it is strictly so.)
1. [5 marks] (SF) A = {(x, y, z) ∈ R3 : x ≥ 0, y ≥ 0 and z > 0}. f(x, y, z) = xy+log z
for every (x, y, z) ∈ A.
2. [5 marks] (SF) A = {(x, y, z) ∈ R3 : x2 + y2 + z2 ≤ 100 and 3x + 2y + 7z ≥ −1}.
f(x, y, z) = exp(x+ y − z) for every (x, y, z) ∈ A.
3. [5 marks] (SF) A = {(x, y, z) ∈ R3 : x2 + y2 + z2 ≥ 100 and 3x + 2y + 7z ≤ −1}.
f(x, y, z) = log(x2 + y2 + 15z2) for every (x, y, z) ∈ A.
Page 3 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
Question 3 [10 marks] (SF) Solve the following consumer’s problem:
maxx1,x2,x3 x1
s.t. xi > 0 for i = 1, 2, 3;
p1x1 + p2x2 + p3x3 ≤ y.
Here y, p1, p2 and p3 are treated as known positive numbers. (Hint: we have shown in
a tutorial that the objective function is strictly quasi-concave; you may use that result
without proof. It is so easy to show that the feasible set is a convex set that you may also
assume that to save some time.)
Page 4 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
Question 4 [10 marks] Consider the following two functions on R:
f0(z) = − log(exp(z) + 1), for every z ∈ R;
f1(z) = z − log(exp(z) + 1), for every z ∈ R.
1. [5 marks] (SF) Find the limits of f0 and f1 when z → ∞ (or show that they do not
2. [5 marks] (SF) Find the limits of f0 and f1 when z → −∞ (or show that they do
not exist).
This is largely an EMET7001 exercise, but you are assumed to have mastered the prereq-
uisite of this course!
Page 5 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
Question 5 [20 marks] Consider the following optimization problem.
[ytxtβ − log(exp(xtβ) + 1)]. (1)
For the purpose of this question, each yt is treated as a known number equal to either 0
or 1, and each xt is a known 1× n matrix. Assume that there exists no β ∈ Rn such that
xtβ = 0 for every t.
1. [5 marks] (SF) Prove the following result: for every v ∈ Rn, lims→∞ |xt(sv)| = ∞
for at least one t.
2. [5 marks] (Medium) Suppose that yt = 1 and every entry of xt is non-negative for
every t. Does the problem Eq. (1) have a solution?
3. [10 marks] (Challenging) Show that the problem Eq.(1) has a solution when the
following condition holds: for every v ∈ Rn, (2yt − 1)xtv is negative for some other
t. (Which (2yt − 1)xtv is negative may depend on v.)
Page 6 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
Question 6 [30 marks] A producer uses n inputs to produce an output. Denote the
quantity of input i by xi and the output by y; the unit price of input i is pi and the unit
price of the output is s. The prices s and all pi’s are positive known numbers. With inputs
(x1, x2, …, xn), the producer’s technology allows her to produce output f(x1, …, xn), where
f : Rn → R is a known twice continuously differentiable and strictly concave function. We
also assume that f is strictly increasing in each of its variables. The producer’s problem
is as follows.
π(s, p) = maxy,x sy − p · x; (2)
s.t. xi ≥ 0, for i = 1, …, n; (3)
y ≥ 0; (4)
y ≤ f(x). (5)
Here x = (x1, …, xn) and p = (p1, …, pn); of course, p · x =
1. [10 marks] (Medium) Can you show that the producer’s problem always has a solu-
2. [5 marks] (Medium) If it can be proved that the problem always has a solution,
great. Otherwise, it will be assumed that additional conditions have been imposed
to ensure the existence of solution for the rest of the question. Now show that the
producer’s problem has a unique solution. (Hint: it would be unreasonable to ask
you to produce such a proof from scratch. Actually, we did a similar proof in a
3. [10 marks] (Challenging) Show that the optimal quantity of Input i, x∗i (s, p), is
weakly decreasing in pi. This is true for every i.
4. [5 marks] (Medium) Assume the conclusion of the previous part even if you cannot
prove it. Is it ever possible that x∗i (s, p) is NOT strictly decreasing in pi?
——— End of Examination ———
———————————————
————————————
Page 7 of 7 – OPTIMIZATION FOR ECONOMICS AND FINANCIAL ECONOMICS (ECON 8013)
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com