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ECON10192/20192 INTRODUCTION TO MATHEMATICAL ECONOMICS 2020-21
发布时间：2022-05-17
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ECON10192/20192
INTRODUCTION TO MATHEMATICAL ECONOMICS
2020-21
Part A: Please answer any TWO of Questions 1-3.
1. (a) What is a sequence? Describe (formally and in your own words) what it means for such a sequence to converge to a limit z 2 R.
(b) Compute the first five elements of the sequence {zn}
1 , where
2n2
zn =
(c) Prove the sequences {
n}
1 and {yn}
1 converge, where
2n 2n
ƒ(
) = 
Plot this function in a graph .
(c) Prove that ƒ is C1 at 
0 = 1 .
(d) Prove whether ƒ is Ck at 
0 = 1 for any k > 1 . Is it Ck for all k > 1 everywhere else? Explain your answer.
3. (a) Define what it means for a set U ✓ R to be compact . (b) Prove that the set given by U = [0, 1) [ [2, 3] is not compact .
g(
) = m
x ¶1 − (
− 1)2, 
− 1 © .
Plot this function in a graph .
(d) In your own words, explain what it means for 
0 2 U to be a global maximum of the function g . Prove that g does not have a global maximum . Discuss this fact in the light of your answer to part (b) .
Part B: Please answer ALL of Question 4.
4. A professor chooses how much time to spend on three activities: re- search, teaching, and administrative duties . The amount of time spent conducting research is written 
, the amount of time spent teaching y, and the amount of time spent attending to administrative duties z . The professor has a total amount of time t > 1, and has preferences over these three activities represented by a utility function 
: R 
7! R, with
(
,y,z) = ln 
+ lny + 
lnz, where 
> 0
is a fixed parameter. The time spent conducting research must be at least 1 or else the professor will be fired; y and z cannot be negative .
(a) Write down the professor’s constrained optimization problem . [4]
(b) Prove that the professor’s objective function is concave and that [8] the professor’s constraint set is convex .
(c) Why will a non-degenerate constraint qualification (NDCQ) hold? [2]
(d) Write down the Lagrangian L for this problem . Compute the first- [4] order conditions . Write down the associated complementary slack- ness, feasibility, and non-negative multiplier conditions .
(e) Briefly explain why the solution to the equations in your answer to [2] part (d) will generate the solution to the problem in part (a) .
(f) Find the optimal amount of time spent on each activity (
∗,y∗,z∗ ) [8] when t > 
+ 2 . What happens when t 
+ 2?
(g) Derive the marginal utility of time as a function of t and plot it in a [8] graph . Is this function continuous at t = 
+ 2? Is it differentiable?
(h) Show that the multiplier associated with the constraint that at least [4] time 1 must be spent conducting research is given by
m
x ⇢0, 
1(1) − 1 
.
Interpret this multiplier both when t > 
+ 2 and when t 
+ 2 .
