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ECON10192/20192 INTRODUCTION TO MATHEMATICAL ECONOMICS 2021-22
发布时间：2022-05-17
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ECON10192/20192
INTRODUCTION TO MATHEMATICAL ECONOMICS
2021-22
Please answer any TWO of Questions 1-3.
1. [30 MARKS] Let z be the month of your birthday.1 Answer each of the following ques- tions:
(a) [5 MARKS] Find the limit of the sequences {xn }
and {yn }
ln(1 + z)n z2
1+(z +1)n (z +1)n
(b) [5 MARKS] Using your answer from (a), find the limit of zn = 
⇣1+ 
⌘
(c) [7 MARKS] Prove whether the function g(x,y) = |(z + 1)(x−y) − 1| is continuous on R2 .
(d) [7 MARKS] State the properties that a function must satisfy to be a distance func- tion. Is the function g(x,y) defined above a distance function? Prove or disprove.
(e) [6 MARKS] Prove the following statement: Let ↵ > 0 and consider any function
f : R ! R satisfying the property |f(x) − f(y)| 
↵|x − y| for all x,y, 2 R. Show that f is continuous on R.
2. [30 MARKS] Let z be the month of your birthday. Answer each of the following questions:
(a) [6 MARKS] Write down a zth order Taylor approximation of the function g(x) = xz
around x = 1. Explain intuitively what the remainder of the polynomial should be.
(b) [6 MARKS] Formally define a quasi-concave function. Prove whether g(x) is a
quasi-concave function.
(c) [8 MARKS] Suppose f is di↵erentiable on X. Prove the following statement using the Mean Value Theorem: If 
≥ 0 for all x 2 X, then f is increasing on X.
(d) [10 MARKS] Consider the system of equations
x2 − y2 − u3 + v2 = pz
2xy + z2 − 2u2 +3v4 = −8
Prove whether you can write u and v in terms of x and y around the point (x,y,u,v) = (2, −1, 2, 1). If yes, calculate the derivative 
.
3. [30 MARKS] Let z be the month of your birthday. Consider the subset X = [−1,z) [ (z,13] of R and answer each of the following questions:
(a) [8 MARKS] Define the real-valued function f : X ! R as
f(x) = 
Draw this function in a graph and explain formally whether f is continuous on X.
(b) [6 MARKS] Explain whether the set X is compact.
(c) [8 MARKS] Define what it means for a function to be di↵erentiable at x 2 X. Identify points in X where f is not di↵erentiable. Provide formal arguments.
(d) [8 MARKS] Can you guarantee a maximum and minimum of f on X? Find the global maxima and minima, if any.
Please answer ALL parts of Question 4.
4. [40 MARKS] Let z be the month of your birthday. Consider a firm with a C2 production technology f(k,l) = k2 + l2 . It faces prices of 6 and z for inputs k and l respectively. The firm has a capacity constraint of 1 on each input, i.e. the firm can use at-most 1 unit of each input. Inputs cannot be used in negative amounts and the firm produces at-least 
units.
(a) [4 MARKS] Write down the cost minimisation problem of the firm by converting it
into a maximisation of the negative cost function.
(b) [5 MARKS] Draw the constraint set in a graph and prove whether it is a compact
set.
(c) [4 MARKS] Suppose 
= 2. Write down the constraint set and explain whether this set is open or closed in R2 .
For the rest of the question, assume 
= 1 and the prices that the firms face for k and l are 6 and z respectively.
(d) [7 MARKS] Argue that we can ignore the constraints k 
1 and l 
1 in the optimisation problem.
(e) [10 MARKS] Write down the Lagrange function and find all its critical points.
(f) [5 MARKS] Calculate the amount by which optimal cost for the firm will rise if 
is
doubled from the original value.
(g) [5 MARKS] Intuitively explain how your answer would change if the capacity con-
straint for capital is reduced to 0.5.
