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ECON20120/30320 Mathematical Economics I EXAM 2022/23
发布时间：2023-01-28
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ECON20120/30320
Mathematical Economics I
EXAM 2022/23 (term 1)
Questions. Answer 3 out of 4.
Instructions
This part contains several questions to be completed before the dead- line. Your solution has to be uploaded to Blackboard. Your answer has to be typed (not hand-written!). If you would like to add figure or graphs, you are permitted to draw these by hand and paste into your answer. Please submit one single PDF document on Blackboard.
You should carefully explain the reasoning behind your answers and calculations. Be advised that the examiners attach considerable im- portance to the clarity with which answers are expressed. Show all the steps of your argument or calculation in your answer.
Answer 3 out of 4 questions. The first 3 questions in your submission will be marked. If you submit solutions to all 4 questions, the last one on your submission will not be looked at. All questions have equal weight (one third of 100 = 33 
).
Question 1
Let X c R × R. For any two bundles x = (x1 , x2 ) e X and y = (y1 , y2 ) e X define the relation 5L as follows:
x 5L y if and only if [x1 > y1] or [x1 = y1 and x2 < y2].
(1) Assume the choice space is X = [0, 1] × Q. Can this preference relationship be represented by a utility function? Show all details of your claim and its proof.
(2) Assume the choice space is X = N × [_1, 1]. Can this preference relationship be represented by a utility function? Show all details of your claim and its proof.
Question 2
Consider the pollution1 minimization problem
M (w, m) = min P (x)
北
s.t. w1 . x1 + ... + wn . xn < m
where P : Rn - [0, o[, w1 , ..., wn are the prices of cleaning products and m is the budget.
Prove that the function M (w, m) : R
× R+ - R is quasi-concave.
[Hint: the subscript “+” means that all elements of a vector are non-negative and at least one element is strictly larger than zero.]
Question 3
Show that the following function
┌ 
← 
dK┐
can be approximated by
_ 
/ 
_ 1、2
where T > 0 is time, r > 0 is a net interest rate, K0 and K are strike prices, and F0 is a futures price. putK and callK are put resp. call options with strike price K that expire time T in the future.
Use the following information:
(1) The put-call parity is given by:
putK = callK + e …rT K _ S0 .
(2) The future price F0 of the underlying and its current price are related as:
F0 = erT . S0 .
(3) K0 ≈ F0 . Hint: this assumption allows the application of a Taylor approximation (see the latest Handout on Blackboard, Section 13.6).
Question 4
Consider a project that runs over n + 1 periods with payments x0 , x1 , ...., xn . The present value at some interest rate r > 0 is
PV (x, r) = i
0 / 
、i xi .
Define the external rate of return rО (x) as the largest(!) interest rate r > 0 such that PV (x, r) = 0.
(1) Find conditions on the set of projects X c Rn+1 such that the external rate of return rО (x) exists and is unique.
(2) Find conditions on the set of projects X c Rn+1 such that the function
r О : X - R
x -1 r О (x)
is quasi-concave.
Instructions: In (1) and (2), your conditions have to be as general as possible. Once you have written down some conditions and proved that these are sufficient, illustrate why it is difficult to make these conditions more general (e.g., with counterexamples). Show all the steps of your argument or reasoning! The conditions in (2) are very likely more restrictive than those you need for (1).
